ELECTROMAGNETICS
VOLUME 1
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ELECTROMAGNETICS
STEVEN W. ELLINGSON
VOLUME 1
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1. Electromagnetism. 2. Electromagnetic theory.
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Contents
Preface
xii
1
Preliminary Concepts
1
1.1
What is Electromagnetics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Fundamentals of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Guided and Unguided Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.5
Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.6
Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2
Electric and Magnetic Fields
17
2.1
What is a Field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2
Electric Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3
Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4
Electric Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5
Magnetic Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6
Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7
Magnetic Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8
Electromagnetic Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
vi
CONTENTS
vii
3 Transmission Lines
30
3.1
Introduction to Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2
Types of Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3
Transmission Lines as Two-Port Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4
Lumped-Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5
Telegrapher’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6
Wave Equation for a TEM Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7
Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.8
Wave Propagation on a TEM Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.9
Lossless and Low-Loss Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.10 Coaxial Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.11 Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.12 Voltage Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.13 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.14 Standing Wave Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15 Input Impedance of a Terminated Lossless Transmission Line . . . . . . . . . . . . . . . . . . . 52
3.16 Input Impedance for Open- and Short-Circuit Terminations . . . . . . . . . . . . . . . . . . . . 54
3.17 Applications of Open- and Short-Circuited Transmission Line Stubs . . . . . . . . . . . . . . . 55
3.18 Measurement of Transmission Line Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 56
3.19 Quarter-Wavelength Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.20 Power Flow on Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.21 Impedance Matching: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.22 Single-Reactance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.23 Single-Stub Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Vector Analysis
70
4.1
Vector Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2
Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
CONTENTS
viii
4.3
Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4
Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5
Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6
Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.7
Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.8
Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9
Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10 The Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5
Electrostatics
93
5.1
Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2
Electric Field Due to Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3
Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4
Electric Field Due to a Continuous Distribution of Charge . . . . . . . . . . . . . . . . . . . . . 96
5.5
Gauss’ Law: Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6
Electric Field Due to an Infinite Line Charge using Gauss’ Law . . . . . . . . . . . . . . . . . . 101
5.7
Gauss’ Law: Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8
Force, Energy, and Potential Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.9
Independence of Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.10 Kirchoff’s Voltage Law for Electrostatics: Integral Form . . . . . . . . . . . . . . . . . . . . . 108
5.11 Kirchoff’s Voltage Law for Electrostatics: Differential Form . . . . . . . . . . . . . . . . . . . 109
5.12 Electric Potential Field Due to Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.13 Electric Potential Field due to a Continuous Distribution of Charge . . . . . . . . . . . . . . . . 112
5.14 Electric Field as the Gradient of Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.15 Poisson’s and Laplace’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.16 Potential Field Within a Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.17 Boundary Conditions on the Electric Field Intensity (E) . . . . . . . . . . . . . . . . . . . . . . 118
5.18 Boundary Conditions on the Electric Flux Density (D) . . . . . . . . . . . . . . . . . . . . . . 120
CONTENTS
ix
5.19 Charge and Electric Field for a Perfectly Conducting Region . . . . . . . . . . . . . . . . . . . 122
5.20 Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.21 Dielectric Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.22 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.23 The Thin Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.24 Capacitance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.25 Electrostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Steady Current and Conductivity
134
6.1
Convection and Conduction Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2
Current Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3
Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4
Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.5
Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6
Power Dissipation in Conducting Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7 Magnetostatics
146
7.1
Comparison of Electrostatics and Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2
Gauss’ Law for Magnetic Fields: Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.3
Gauss’ Law for Magnetism: Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4
Ampere’s Circuital Law (Magnetostatics): Integral Form . . . . . . . . . . . . . . . . . . . . . 149
7.5
Magnetic Field of an Infinitely-Long Straight Current-Bearing Wire . . . . . . . . . . . . . . . 150
7.6
Magnetic Field Inside a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.7
Magnetic Field of a Toroidal Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.8
Magnetic Field of an Infinite Current Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.9
Ampere’s Law (Magnetostatics): Differential Form . . . . . . . . . . . . . . . . . . . . . . . . 159
7.10 Boundary Conditions on the Magnetic Flux Density (B) . . . . . . . . . . . . . . . . . . . . . . 160
7.11 Boundary Conditions on the Magnetic Field Intensity (H) . . . . . . . . . . . . . . . . . . . . . 161
CONTENTS
x
7.12 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.13 Inductance of a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.14 Inductance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.15 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.16 Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8
9
Time-Varying Fields
174
8.1
Comparison of Static and Time-Varying Electromagnetics . . . . . . . . . . . . . . . . . . . . 174
8.2
Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3
Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.4
Induction in a Motionless Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.5
Transformers: Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.6
Transformers as Two-Port Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.7
The Electric Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.8
The Maxwell-Faraday Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.9
Displacement Current and Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Plane Waves in Lossless Media
194
9.1
Maxwell’s Equations in Differential Phasor Form . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.2
Wave Equations for Source-Free and Lossless Regions . . . . . . . . . . . . . . . . . . . . . . 196
9.3
Types of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.4
Uniform Plane Waves: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.5
Uniform Plane Waves: Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.6
Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.7
Wave Power in a Lossless Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
A Constitutive Parameters of Some Common Materials
213
A.1 Permittivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.2 Permeability of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
CONTENTS
xi
A.3 Conductivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B Mathematical Formulas
217
B.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.3 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
C Physical Constants
220
Index
221
Preface
About This Book
• Accessibility features: Figures now include “alt
text” suitable for screen reading software.
• Addition of a separate manual of examples and
solutions (see the web site).
[m0146]
Goals for this book. This book is intended to serve
as a primary textbook for a one-semester introductory
course in undergraduate engineering
electromagnetics, including the following topics:
electric and magnetic fields; electromagnetic
properties of materials; electromagnetic waves; and
devices that operate according to associated
electromagnetic principles including resistors,
capacitors, inductors, transformers, generators, and
transmission lines.
This book employs the “transmission lines first”
approach, in which transmission lines are introduced
using a lumped-element equivalent circuit model for a
differential length of transmission line, leading to
one-dimensional wave equations for voltage and
current.1 This is sufficient to address transmission
line concepts, including characteristic impedance,
input impedance of terminated transmission lines, and
impedance matching techniques. Attention then turns
to electrostatics, magnetostatics, time-varying fields,
and waves, in that order.
What’s new. This version of the book is the second
public release of this book. The first release, known
as “Volume 1 (Beta),” was released in January 2018.
Improvements from the beta version include the
following:
• Correction of errors identified in the beta version
errata and many minor improvements.
• Addition of an index.
1 Are you an instructor who is not a fan of the “transmission
lines first” approach? Then see “What are those little numbers
in square brackets?” later in this section.
• Addition of source files for the book (see the
web site).
Target audience. This book is intended for electrical
engineering students in the third year of a bachelor of
science degree program. It is assumed that readers are
familiar with the fundamentals of electric circuits and
linear systems, which are normally taught in the
second year of the degree program. It is also assumed
that readers have received training in basic
engineering mathematics, including complex
numbers, trigonometry, vectors, partial differential
equations, and multivariate calculus. Review of the
relevant principles is provided at various points in the
text. In a few cases (e.g., phasors, vectors) this review
consists of a separate stand-alone section.
Notation, examples, and highlights. Section 1.7
summarizes the mathematical notation used in this
book. Examples are set apart from the main text as
follows:
Example 0.1. This is an example.
“Highlight boxes” are used to identify key ideas as
follows:
This is a key idea.
What are those little numbers in square brackets?
This book is a product of the
Open Electromagnetics Project. This project provides
a large number of sections (“modules”) which are
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
xiii
assembled (“remixed”) to create new and different
versions of the book. The text “[m0146]” that you see at
the beginning of this section uniquely identifies the
module within the larger set of modules provided by
the project. This identification is provided because
different remixes of this book may exist, each
consisting of a different subset and arrangement of
these modules. Prospective authors can use this
identification as an aid in creating their own remixes.
Why do some sections of this book seem to repeat
material presented in previous sections? In some
remixes of this book, authors might choose to
eliminate or reorder modules. For this reason, the
modules are written to “stand alone” as much as
possible. As a result, there may be some redundancy
between sections that would not be present in a
traditional (non-remixable) textbook. While this may
seem awkward to some at first, there are clear
benefits: In particular, it never hurts to review relevant
past material before tackling a new concept. And,
since the electronic version of this book is being
offered at no cost, there is not much gained by
eliminating this useful redundancy.
Why cite Wikipedia pages as additional reading?
Many modules cite Wikipedia entries as sources of
additional information. Wikipedia represents both the
best and worst that the Internet has to offer. Most
authors of traditional textbooks would agree that
citing Wikipedia pages as primary sources is a bad
idea, since quality is variable and content is subject to
change over time. On the other hand, many Wikipedia
pages are excellent, and serve as useful sources of
relevant information that is not strictly within the
scope of the curriculum. Furthermore, students
benefit from seeing the same material presented
differently, in a broader context, and with additional
references cited by Wikipedia pages. We trust
instructors and students to realize the potential pitfalls
of this type of resource and to be alert for problems.
Acknowledgments. Here’s a list of talented and
helpful people who contributed to this book:
The staff of VT Publishing, University Libraries,
Virginia Tech:
Editor: Anita Walz
Advisors: Peter Potter, Corinne Guimont
Cover: Robert Browder, Anita Walz
Other VT contributors:
Assessment: Tiffany Shoop, Anita Walz
Accessibility: Christa Miller
Virginia Tech students:
Alt text writer: Stephanie Edwards
Figure designer: Michaela Goldammer
Figure designer: Kruthika Kikkeri
Figure designer: Youmin Qin
Copyediting:
Melissa Ashman, Kwantlen Polytechnic University
External reviewers:
Samir El-Ghazaly, University of Arkansas
Stephen Gedney, University of Colorado Denver
Randy Haupt, Colorado School of Mines
Karl Warnick, Brigham Young University
Also, thanks are due to the students of the Spring
2018 and Summer 2018 sections of ECE3105 at
Virginia Tech who used the beta version of this book
and provided useful feedback. Thanks also to Justin
Yonker, instructor of the Summer 2018 section.
PREFACE
xiv
About the Open Electromagnetics
Project
About the Author
[m0153]
[m0148]
The Open Electromagnetics Project was established
at Virginia Tech in 2017 with the goal of creating
no-cost openly-licensed textbooks for courses in
undergraduate engineering electromagnetics. While a
number of very fine traditional textbooks are available
on this topic, we feel that it has become unreasonable
to insist that students pay hundreds of dollars per
book when effective alternatives can be provided
using modern media at little or no cost to the student.
This project is equally motivated by the desire for the
freedom to adopt, modify, and improve educational
resources. This work is distributed under a Creative
Commons BY SA license which allows – and we
hope encourages – others to adopt, modify, improve,
and expand the scope of our work.
Steven W. Ellingson (ellingson@vt.edu) is an
Associate Professor at Virginia Tech in Blacksburg,
Virginia in the United States. He received PhD and
MS degrees in Electrical Engineering from the Ohio
State University and a BS in Electrical & Computer
Engineering from Clarkson University. He was
employed by the US Army, Booz-Allen & Hamilton,
Raytheon, and the Ohio State University
ElectroScience Laboratory before joining the faculty
of Virginia Tech, where he teaches courses in
electromagnetics, radio frequency electronics,
wireless communications, and signal processing. His
research includes topics in wireless communications,
radio science, and radio frequency instrumentation.
Professor Ellingson serves as a consultant to industry
and government and is the author of Radio Systems
Engineering (Cambridge University Press, 2016).
Chapter 1
Preliminary Concepts
1.1 What is Electromagnetics?
[m0037]
The topic of this book is applied engineering
electromagnetics. This topic is often described as “the
theory of electromagnetic fields and waves,” which is
both true and misleading. The truth is that electric
fields, magnetic fields, their sources, waves, and the
behavior these waves are all topics covered by this
book. The misleading part is that our principal aim
shall be to close the gap between basic electrical
circuit theory and the more general theory that is
required to address certain topics that are of broad and
common interest in the field of electrical engineering.
(For a preview of topics where these techniques are
required, see the list at the end of this section.)
In basic electrical circuit theory, the behavior of
devices and systems is abstracted in such a way that
the underlying electromagnetic principles do not need
to be considered. Every student of electrical
engineering encounters this, and is grateful since this
greatly simplifies analysis and design. For example, a
resistor is commonly defined as a device which
exhibits a particular voltage V = IR in response to a
current I, and the resistor is therefore completely
described by the value R. This is an example of a
“lumped element” abstraction of an electrical device.
Much can be accomplished knowing nothing else
about resistors; no particular knowledge of the
physical concepts of electrical potential, conduction
current, or resistance is required. However, this
simplification makes it impossible to answer some
frequently-encountered questions. Here are just a few:
• What determines R? How does one go about
designing a resistor to have a particular
resistance?
• Practical resistors are rated for power-handling
capability; e.g., discrete resistors are frequently
identified as “1/8-W,” “1/4-W,” and so on. How
does one determine this, and how can this be
adjusted in the design?
• Practical resistors exhibit significant reactance as
well as resistance. Why? How is this
determined? What can be done to mitigate this?
• Most things which are not resistors also exhibit
significant resistance and reactance – for
example, electrical pins and interconnects. Why?
How is this determined? What can be done to
mitigate this?
The answers to the these questions must involve
properties of materials and the geometry in which
those materials are arranged. These are precisely the
things that disappear in lumped element device
models, so it is not surprising that such models leave
us in the dark on these issues. It should also be
apparent that what is true for the resistor is also going
to be true for other devices of practical interest,
including capacitors (and devices unintentionally
exhibiting capacitance), inductors (and devices
unintentionally exhibiting inductance), transformers
(and devices unintentionally exhibiting mutual
impedance), and so on. From this perspective,
electromagnetics may be viewed as a generalization
of electrical circuit theory that addresses these
considerations. Conversely basic electric circuit
theory may be viewed a special case of
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
CHAPTER 1. PRELIMINARY CONCEPTS
2
electromagnetic theory that applies when these
considerations are not important. Many instances of
this “electromagnetics as generalization” vs.
“lumped-element theory as special case” dichotomy
appear in the study of electromagnetics.
There is more to the topic, however. There are many
devices and applications in which electromagnetic
fields and waves are primary engineering
considerations that must be dealt with directly.
Examples include electrical generators and motors;
antennas; printed circuit board stackup and layout;
persistent storage of data (e.g., hard drives); fiber
optics; and systems for radio, radar, remote sensing,
and medical imaging. Considerations such as signal
integrity and electromagnetic compatibility (EMC)
similarly require explicit consideration of
electromagnetic principles.
Although electromagnetic considerations pertain to
all frequencies, these considerations become
increasingly difficult to avoid with increasing
frequency. This is because the wavelength of an
electromagnetic field decreases with increasing
frequency.1 When wavelength is large compared to
the size of the region of interest (e.g., a circuit), then
analysis and design is not much different from
zero-frequency (“DC”) analysis and design.
For example, the free space wavelength at 3 MHz is
about 100 m, so a planar circuit having dimensions
10 cm × 10 cm is just 0.1% of a wavelength across at
this frequency. Although an electromagnetic wave
may be present, it has about the same value over the
region of space occupied by the circuit. In contrast,
the free space wavelength at 3 GHz is about 10 cm, so
the same circuit is one full wavelength across at this
frequency. In this case, different parts of this circuit
observe the same signal with very different magnitude
and phase.
Some of the behaviors associated with non-negligible
dimensions are undesirable, especially if not taken
into account in the design process. However, these
behaviors can also be exploited to do some amazing
and useful things – for example, to launch an
electromagnetic wave (i.e., an antenna) or to create
1 Most readers have encountered the concepts of frequency and
wavelength previously, but can refer to Section 1.3, if needed, for a
quick primer.
filters and impedance matching devices consisting
only of metallic shapes, free of discrete capacitors or
inductors.
Electromagnetic considerations become not only
unavoidable but central to analysis and design above a
few hundred MHz, and especially in the
millimeter-wave, infrared (IR), optical, and ultraviolet
(UV) bands.2 The discipline of electrical engineering
encompasses applications in these frequency ranges
even though – ironically – such applications may not
operate according to principles that can be considered
“electrical”! Nevertheless, electromagnetic theory
applies.
Another common way to answer the question “What
is electromagnetics?” is to identify the topics that are
commonly addressed within this discipline. Here’s a
list of topics – some of which have already been
mentioned – in which explicit consideration of
electromagnetic principles is either important or
essential:3
• Antennas
• Coaxial cable
• Design and characterization of common discrete
passive components including resistors,
capacitors, inductors, and diodes
• Distributed (e.g., microstrip) filters
• Electromagnetic compatibility (EMC)
• Fiber optics
• Generators
• Magnetic resonance imaging (MRI)
• Magnetic storage (of data)
• Microstrip transmission lines
• Modeling of non-ideal behaviors of discrete
components
• Motors
2 See Section 1.2 for a quick primer on the electromagnetic spectrum and this terminology.
3 Presented in alphabetical order so as to avoid the appearance of
any bias on the part of the author!
1.2. ELECTROMAGNETIC SPECTRUM
• Non-contact sensors
• Photonics
• Printed circuit board stackup and layout
• Radar
• Radio wave propagation
• Radio frequency electronics
• Signal integrity
• Transformers
• Waveguides
In summary:
Applied engineering electromagnetics is the
study of those aspects of electrical engineering in
situations in which the electromagnetic properties of materials and the geometry in which those
materials are arranged is important. This requires an understanding of electromagnetic fields
and waves, which are of primary interest in some
applications.
Finally, here are two broadly-defined learning
objectives that should now be apparent: (1) Learn the
techniques of engineering analysis and design that
apply when electromagnetic principles are important,
and (2) Better understand the physics underlying the
operation of electrical devices and systems, so that
when issues associated with these physical principles
emerge one is prepared to recognize and grapple with
them.
3
1.2 Electromagnetic Spectrum
[m0075]
Electromagnetic fields exist at frequencies from DC
(0 Hz) to at least 1020 Hz – that’s at least 20 orders of
magnitude!
At DC, electromagnetics consists of two distinct
disciplines: electrostatics, concerned with electric
fields; and magnetostatics, concerned with magnetic
fields.
At higher frequencies, electric and magnetic fields
interact to form propagating waves. Waves having
frequencies within certain ranges are given names
based on how they manifest as physical phenomena.
These names are (in order of increasing frequency):
radio, infrared (IR), optical (also known as “light”),
ultraviolet (UV), X-rays, and gamma rays (γ-rays).
See Table 1.1 and Figure 1.1 for frequency ranges and
associated wavelengths.
The term electromagnetic spectrum refers to the
various forms of electromagnetic phenomena that
exist over the continuum of frequencies.
The speed (properly known as “phase velocity”) at
which electromagnetic fields propagate in free space
is given the symbol c, and has the value
∼
= 3.00 × 108 m/s. This value is often referred to as
the “speed of light.” While it is certainly the speed of
light in free space, it is also speed of any
electromagnetic wave in free space. Given frequency
f , wavelength is given by the expression
λ=
c
in free space
f
Table 1.1 shows the free space wavelengths
associated with each of the regions of the
electromagnetic spectrum.
This book presents a version of electromagnetic
theory that is based on classical physics. This
approach works well for most practical problems.
However, at very high frequencies, wavelengths
become small enough that quantum mechanical
effects may be important. This is usually the case in
the X-ray band and above. In some applications, these
CHAPTER 1. PRELIMINARY CONCEPTS
4
Regime
γ-Ray
X-Ray
Ultraviolet (UV)
Optical
Infrared (IR)
Radio
Frequency Range
> 3 × 1019 Hz
3 × 1016 Hz – 3 × 1019 Hz
2.5 × 1015 – 3 × 1016 Hz
4.3 × 1014 – 2.5 × 1015 Hz
300 GHz – 4.3 × 1014 Hz
3 kHz – 300 GHz
Wavelength Range
< 0.01 nm
10–0.01 nm
120–10 nm
700–120 nm
1 mm – 700 nm
100 km – 1 mm
Table 1.1: The electromagnetic spectrum. Note that the indicated ranges are arbitrary but consistent with
common usage.
effects become important at frequencies as low as the
optical, IR, or radio bands. (A prime example is the
photoelectric effect; see “Additional References”
below.) Thus, caution is required when applying the
classical version of electromagnetic theory presented
here, especially at these higher frequencies.
Wavelength
Frequency (Hz)
Gamma-rays
0.1 Å
1019
1Å
0.1 nm
1018
Theory presented in this book is applicable to
DC, radio, IR, and optical waves, and to a lesser
extent to UV waves, X-rays, and γ-rays. Certain phenomena in these frequency ranges – in
particular quantum mechanical effects – are not
addressed in this book.
400 nm
X-rays
1 nm
1017
10 nm
1016
500 nm
Ultraviolet
100 nm
10
15
Visible
Near IR
10
1000 nm
1 µm
14
The radio portion of the electromagnetic spectrum
alone spans 12 orders of magnitude in frequency (and
wavelength), and so, not surprisingly, exhibits a broad
range of phenomena. This is shown in Figure 1.1. For
this reason, the radio spectrum is further subdivided
into bands as shown in Table 1.2. Also shown in
Table 1.2 are commonly-used band identification
acronyms and some typical applications.
Similarly, the optical band is partitioned into the
familiar “rainbow” of red through violet, as shown in
Figure 1.1 and Table 1.3. Other portions of the
spectrum are sometimes similarly subdivided in
certain applications.
1013
Thermal IR
100 µm
1012
Far IR
1000 MHz
1011
UHF
Microwaves
1010
500 MHz
1000 µm
1 mm
1 cm
Radar
10 cm
109
VHF
7-13
100 MHz
Radio, TV
FM
VHF
2-6
50 MHz
1m
108
10 m
107
100 m
106
AM
1000 m
Long-waves
Additional Reading:
c V. Blacus CC BY-SA 3.0
• “Electromagnetic spectrum” on Wikipedia.
• “Photoelectric effect” on Wikipedia.
600 nm
10 µm
Infra-red
Figure 1.1: Electromagnetic Spectrum.
700 nm
1.3. FUNDAMENTALS OF WAVES
Band
EHF
SHF
UHF
VHF
HF
MF
LF
VLF
Frequencies
30-300 GHz
3–30 GHz
300–3000 MHz
30–300 MHz
3–30 MHz
300–3000 kHz
30–300 kHz
3–30 kHz
5
Wavelengths
10–1 mm
10–1 cm
1–0.1 m
10–1 m
100–10 m
1000–100 m
10–1 km
100–10 km
Typical Applications
60 GHz WLAN, Point-to-point data links
Terrestrial & Satellite data links, Radar
TV broadcasting, Cellular, WLAN
FM & TV broadcasting, LMR
Global terrestrial comm., CB Radio
AM broadcasting
Navigation, RFID
Navigation
Table 1.2: The radio portion of the electromagnetic spectrum, according to a common scheme for naming ranges
of radio frequencies. WLAN: Wireless local area network, LMR: Land mobile radio, RFID: Radio frequency
identification.
Band
Violet
Blue
Green
Yellow
Orange
Red
Frequencies
668–789 THz
606–668 THz
526–606 THz
508–526 THz
484–508 THz
400–484 THz
Wavelengths
450–380 nm
495–450 nm
570–495 nm
590–570 nm
620–590 nm
750–620 nm
Table 1.3: The optical portion of the electromagnetic
spectrum.
1.3 Fundamentals of Waves
[m0074]
In this section, we formally introduce the concept of a
wave and explain some basic characteristics.
To begin, let us consider not electromagnetic waves,
but rather sound waves. To be clear, sound waves and
electromagnetic waves are completely distinct
phenomena. Sound waves are variations in pressure,
whereas electromagnetic waves are variations in
electric and magnetic fields. However, the
mathematics that govern sound waves and
electromagnetic waves are very similar, so the
analogy provides useful insight. Furthermore, sound
waves are intuitive for most people because they are
readily observed. So, here we go:
Imagine standing in an open field and that it is
completely quiet. In this case, the air pressure
everywhere is about 101 kPa (101,000 N/m2 ) at sea
level, and we refer to this as the quiescent air
pressure. Sound can be described as the differential
air pressure p(x, y, z, t), which we define as the
absolute air pressure at the spatial coordinates
(x, y, z) minus the quiescent air pressure. So, when
there is no sound, p(x, y, z, t) = 0. The function p as
an example of a scalar field.4
Let’s also say you are standing at x = y = z = 0 and
you have brought along a friend who is standing at
x = d; i.e., a distance d from you along the x axis.
Also, for simplicity, let us consider only what is
happening along the x axis; i.e., p(x, t).
At t = 0, you clap your hands once. This forces the
air between your hands to press outward, creating a
region of increased pressure (i.e., p > 0) that travels
outward. As the region of increased pressure moves
outward, it leaves behind a region of low pressure
where p < 0. Air molecules immediately move
toward this region of lower pressure, and so the air
pressure quickly returns to the quiescent value, p = 0.
The traveling disturbance in p(x, t) is the sound of the
clap. The disturbance continues to travel outward
until it reaches your friend, who then hears the clap.
At each point in time, you can make a plot of p(x, t)
versus x for the current value of t. This is shown in
Figure 1.2. At times t < 0, we have simply
p(x, t) = 0. A short time after t = 0, the peak
pressure is located at slightly to the right of x = 0.
The pressure is not a simple impulse because
interactions between air molecules constrain the
4 Although it’s not important in this section, you can read about
the concept of a “field” in Section 2.1.
CHAPTER 1. PRELIMINARY CONCEPTS
6
p(x,t)
blow a perfect A note. The A note is 440 Hz, meaning
that the air pressure emerging from your trumpet is
varying sinusoidally at a frequency of 440 Hz. Let’s
say you can continue to blow this note long enough
for the entire field to be filled with the sound of your
trumpet. Now what does the pressure-versus-distance
curve look like? Two simple observations will settle
that question:
x
p(x,t)
• p(x, t) at any constant position x is a sinusoid as
a function of x. This is because the acoustic
wave equation is linear and time invariant, so a
sinusoidal excitation (i.e., your trumpet) results
in a sinusoidal response at the same frequency
(i.e., the sound heard by your friend).
x
c Y. Qin CC BY 4.0
Figure 1.2: The differential pressure p(x, t) (top) a
short time after the clap and (bottom) a slightly longer
time after the clap.
• p(x, t) at any constant time t is also a sinusoid as
a function of x. This is because the sound is
propagating away from the trumpet and toward
your friend, and anyone in between will also
hear the A note, but with a phase shift
determined by the difference in distances.
pressure to be continuous over space. So instead, we
see a rounded pulse representing the rapid build-up
and similarly rapid decline in air pressure. A short
time later p(x, t) looks very similar, except the pulse
is now further away.
This is enough information to know that the solution
must have the form:
Now: What precisely is p(x, t)? Completely skipping
over the derivation, the answer is that p(x, t) is the
solution to the acoustic wave equation (see
“Additional References” at the end of this section):
1 ∂2p
∂2p
−
=0
∂x2
c2s ∂t2
(1.1)
where cs is the speed of sound, which is about
340 m/s at sea level. Just to emphasize the quality of
the analogy between sound waves and
electromagnetic waves, know that the acoustic wave
equation is mathematically identical to equations that
that govern electromagnetic waves.
Although “transient” phenomena – analogous to a
clap – are of interest in electromagnetics, an even
more common case of interest is the wave resulting
from a sinusoidally-varying source. We can
demonstrate this kind of wave in the context of sound
as well. Here we go:
In the previous scenario, you pick up a trumpet and
p(x, t) = Am cos (ωt − βx + ψ)
(1.2)
where ω = 2πf , f = 440 Hz, and Am , β, and ψ
remain to be determined.
You can readily verify that Equation 1.2 satisfies the
acoustic wave equation when
ω
(1.3)
cs
In this problem, we find β ∼
= 8.13 rad/m. This means
that at any given time, the difference in phase
measured between any two points separated by a
distance of 1 m is 8.13 rad. The parameter β goes by
at least three names: phase propagation constant,
wavenumber, and spatial frequency. The term “spatial
frequency” is particularly apt, since β plays precisely
the same role for distance (x) as ω plays for time (t) –
This is apparent from Equation 1.2. However,
“wavenumber” is probably the more commonly-used
term.
β=
The wavenumber β (rad/m) is the rate at which
the phase of a sinusoidal wave progresses with
distance.
1.3. FUNDAMENTALS OF WAVES
7
Note that Am and ψ are not determined by the wave
equation, but instead are properties of the source.
Specifically, Am is determined by how hard we blow,
and ψ is determined by the time at which we began to
blow and the location of the trumpet. For simplicity,
let us assume that we begin to blow at time t ≪ 0;
i.e., in the distant past so that the sound pressure field
has achieved steady state by t = 0. Also, let us set
ψ = 0 and set Am = 1 in whatever units we choose to
express p(x, t). We then have:
p(x, t) = cos (ωt − βx)
p(x,t)
λ
(1.4)
Now we have everything we need to make plots of
p(x) at various times.
Figure 1.3(a) shows p(x, t = 0). As expected,
p(x, t = 0) is periodic in x. The associated period is
referred to as the wavelength λ. Since λ is the
distance required for the phase of the wave to increase
by 2π rad, and because phase is increasing at a rate of
β rad/m, we find:
2π
(1.5)
λ=
β
In the present example, we find λ ∼
= 77.3 cm.
Wavelength λ = 2π/β is the distance required
for the phase of a sinusoidal wave to increase
by one complete cycle (i.e., 2π rad) at any given
time.
x
p(x,t)
x
p(x,t)
x
Now let us consider the situation at t = +1/4f ,
which is t = 568 µs and ωt = π/2. We see in
Figure 1.3(b) that the waveform has shifted a distance
λ/4 to the right. It is in this sense that we say the
wave is propagating in the +x direction.
Furthermore, we can now compute a phase velocity
vp : We see that a point of constant phase has shifted a
distance λ/4 in time 1/4f , so
vp = λf
(1.6)
∼ 340 m/s; i.e., we
In the present example, we find vp =
have found that the phase velocity is equal to the
speed of sound cs . It is in this sense that we say that
the phase velocity is the speed at which the wave
propagates.5
5 It is worth noting here is that “velocity” is technically a vector;
i.e., speed in a given direction. Nevertheless, this quantity is actually
just a speed, and this particular abuse of terminology is generally
accepted.
c Y. Qin CC BY 4.0
Figure 1.3: The differential pressure p(x, t) for (a)
t = 0, (b) t = 1/4f for “−βx,” as indicated in Equation 1.4 (wave traveling to right); and (c) t = 1/4f for
“+βx” (wave traveling to left).
CHAPTER 1. PRELIMINARY CONCEPTS
8
Phase velocity vp = λf is the speed at which a
point of constant phase in a sinusoidal waveform
travels.
Recall that in Equation 1.2 we declared that βx is
subtracted from the argument of the sinusoidal
function. To understand why, let’s change the sign of
βx and see if it still satisfies the wave equation – one
finds that it does. Next, we repeat the previous
experiment and see what happens. The result is shown
in Figure 1.3(c). Note that points of constant phase
have traveled an equal distance, but now in the −x
direction. In other words, this alternative choice of
sign for βx within the argument of the cosine function
represents a wave that is propagating in the opposite
direction. This leads us to the following realization:
If the phase of the wave is decreasing with βx,
then the wave is propagating in the +x direction.
If the phase of the wave is increasing with βx,
then the wave is propagating in the −x direction.
Since the prospect of sound traveling toward the
trumpet is clearly nonsense in the present situation,
we may neglect the latter possibility. However, what
happens if there is a wall located in the distance,
behind your friend? Then, we expect an echo from
the wall, which would be a second wave propagating
in the reverse direction and for which the argument of
the cosine function would contain the term “+βx.”
Finally, let us return to electromagnetics.
Electromagnetic waves satisfy precisely the same
wave equation (i.e., Equation 1.1) as do sound waves,
except that the phase velocity is much greater.
Interestingly, though, the frequencies of
electromagnetic waves are also much greater than
those of sound waves, so we can end up with
wavelengths having similar orders of magnitude. In
particular, an electromagnetic wave with λ = 77.3 cm
(the wavelength of the “A” note in the preceding
example) lies in the radio portion of the
electromagnetic spectrum.
An important difference between sound and
electromagnetic waves is that electromagnetic waves
are vectors; that is, they have direction as well as
magnitude. Furthermore, we often need to consider
multiple electromagnetic vector waves (in particular,
both the electric field and the magnetic field) in order
to completely understand the situation. Nevertheless
the concepts of wavenumber, wavelength, phase
velocity, and direction of propagation apply in
precisely the same manner to electromagnetic waves
as they do to sound waves.
Additional Reading:
• “Wave Equation” on Wikipedia.
• “Acoustic Wave Equation” on Wikipedia.
1.4. GUIDED AND UNGUIDED WAVES
9
1.4 Guided and Unguided Waves
1.5 Phasors
[m0040]
[m0033]
Broadly speaking, waves may be either guided or
unguided.
In many areas of engineering, signals are
well-modeled as sinusoids. Also, devices that process
these signals are often well-modeled as linear
time-invariant (LTI) systems. The response of an LTI
system to any linear combination of sinusoids is
another linear combination of sinusoids having the
same frequencies.6 In other words, (1) sinusoidal
signals processed by LTI systems remain sinusoids
and are not somehow transformed into square waves
or some other waveform; and (2) we may calculate
the response of the system for one sinusoid at a time,
and then add the results to find the response of the
system when multiple sinusoids are applied
simultaneously. This property of LTI systems is
known as superposition.
Unguided waves include those that are radiated by
antennas, as well as those that are unintentionally
radiated. Once initiated, these waves propagate in an
uncontrolled manner until they are redirected by
scattering or dissipated by losses associated with
materials.
Examples of guided waves are those that exist within
structures such as transmission lines, waveguides, and
optical fibers. We refer to these as guided because
they are constrained to follow the path defined by the
structure.
Antennas and unintentional radiators emit unguided waves. Transmission lines, waveguides,
and optical fibers carry guided waves.
The analysis of systems that process sinusoidal
waveforms is greatly simplified when the sinusoids
are represented as phasors. Here is the key idea:
A phasor is a complex-valued number that represents a real-valued sinusoidal waveform. Specifically, a phasor has the magnitude and phase of
the sinusoid it represents.
Figures 1.4 and 1.5 show some examples of phasors
and the associated sinusoids.
It is important to note that a phasor by itself is not the
signal. A phasor is merely a simplified mathematical
representation in which the actual, real-valued
physical signal is represented as a complex-valued
constant.
Here is a completely general form for a physical
(hence, real-valued) quantity varying sinusoidally
with angular frequency ω = 2πf :
A(t; ω) = Am (ω) cos (ωt + ψ(ω))
(1.7)
where Am (ω) is magnitude at the specified frequency,
ψ(ω) is phase at the specified frequency, and t is time.
Also, we require ∂Am /∂t = 0; that is, that the time
is
6
PA “linear combination” of functions fi (t) where i = 1, 2, 3, ...
i ai fi (t) where the ai ’s are constants.
CHAPTER 1. PRELIMINARY CONCEPTS
10
Im{C}
m
Ame
j
t
ni
ag
e
ud
phase ( )
Re{C}
Am
Am
Bm
t
Am-jBm
-jAm
Ame
jψ
Figure 1.4: Examples of phasors, displayed here as
points in the real-imaginary plane.
variation of A(t) is completely represented by the
cosine function alone. Now we can equivalently
express A(t; ω) as a phasor C(ω):
C(ω) = Am (ω)ejψ(ω)
(1.9)
To see why this works, simply substitute the right
hand side of Equation 1.8 into Equation 1.9. Then:
o
n
A(t) = Re Am (ω)ejψ(ω) ejωt
n
o
= Re Am (ω)ej(ωt+ψ(ω))
= Re {Am (ω) [cos (ωt + ψ(ω))
+j sin (ωt + ψ(ω))]}
= Am (ω) cos (ωt + ψ(ω))
as expected.
It is common to write Equation 1.8 as follows,
dropping the explicit indication of frequency
Am-jBm
note slightly larger amplitude
t
(1.8)
To convert this phasor back to the physical signal it
represents, we (1) restore the time dependence by
multiplying by ejωt , and then (2) take the real part of
the result. In mathematical notation:
A(t; ω) = Re C(ω)ejωt
t
-jAm
note now a sine function
t
Figure 1.5: Sinusoids corresponding to the phasors
shown in Figure 1.4.
1.5. PHASORS
11
dependence:
C = Am ejψ
(1.10)
This does not normally cause any confusion since the
definition of a phasor requires that values of C and ψ
are those that apply at whatever frequency is
represented by the suppressed sinusoidal dependence
ejωt .
and similarly
Re C2 ejωt = Re {C2 · j} = −Im {C2 }
therefore Im {C1 } = Im {C2 }. Once again: Since C1
and C2 are constant with respect to time, this must be
true for all t. Since the real and imaginary parts of C1
and C2 are equal, C1 = C2 .
Table 1.4 shows mathematical representations of the
same phasors demonstrated in Figure 1.4 (and their
associated sinusoidal waveforms in Figure 1.5). It is a
good exercise is to confirm each row in the table,
transforming from left to right and vice-versa.
What does this mean? We have just shown that if two
phasors are equal, then the sinusoidal waveforms that
they represent are also equal.
It is not necessary to use a phasor to represent a
sinusoidal signal. We choose to do so because phasor
representation leads to dramatic simplifications. For
example:
Claim 2: For any real-valued linear operator T and
complex-valued quantity C,
T (Re {C}) = Re {T (C)}.
• Calculation of the peak value from data
representing A(t; ω) requires a time-domain
search over one period of the sinusoid. However,
if you know C, the peak value of A(t) is simply
|C|, and no search is required.
• Calculation of ψ from data representing A(t; ω)
requires correlation (essentially, integration)
over one period of the sinusoid. However, if you
know C, then ψ is simply the phase of C, and no
integration is required.
Furthermore, mathematical operations applied to
A(t; ω) can be equivalently performed as operations
on C, and the latter are typically much easier than the
former. To demonstrate this, we first make two
important claims and show that they are true.
Claim 1: Let C1 and C2 be two complex-valued
constants
(independent
of t). Also,
Re C1 ejωt = Re C2 ejωt for all t. Then,
C1 = C2 .
Proof: Evaluating at t = 0 we find
Re {C1 } = Re {C2 }. Since C1 and C2 are constant
with respect to time, this must be true for all t. At
t = π/(2ω) we find
Re C1 ejωt = Re {C1 · j} = −Im {C1 }
Proof: Let C = cr + jci where cr and ci are
real-valued quantities, and evaluate the right side of
the equation:
Re {T (C)} = Re {T (cr + jci )}
= Re {T (cr ) + jT (ci )}
= T (cr )
= T (Re {C})
What does this mean? The operators that we have in
mind for T include addition, multiplication by a
constant, differentiation, integration, and so on.
Here’s an example with differentiation:
∂
∂
∂
Re
C = Re
(cr + jci ) =
cr
∂ω
∂ω
∂ω
∂
∂
∂
Re {C} =
Re {(cr + jci )} =
cr
∂ω
∂ω
∂ω
In other words, differentiation of a sinusoidal signal
can be accomplished by differentiating the associated
phasor, so there is no need to transform a phasor back
into its associated real-valued signal in order to
perform this operation.
CHAPTER 1. PRELIMINARY CONCEPTS
12
A(t)
Am cos (ωt)
Am cos (ωt + ψ)
Am sin (ωt)
Am cos (ωt) + Bm sin (ωt)
= Am cos ωt − π2
= Am cos (ωt) + Bm cos ωt − π2
C
Am
Am ejψ
−jAm
Am − jBm
Table 1.4: Some examples of physical (real-valued) sinusoidal signals and the corresponding phasors. Am and
Bm are real-valued and constant with respect to t.
Summarizing:
Claims 1 and 2 together entitle us to perform operations on phasors as surrogates for the physical, real-valued, sinusoidal waveforms they represent. Once we are done, we can transform the
resulting phasor back into the physical waveform
it represents using Equation 1.9, if desired.
However, a final transformation back to the time
domain is usually not desired, since the phasor tells us
everything we can know about the corresponding
sinusoid.
A skeptical student might question the value of phasor
analysis on the basis that signals of practical interest
are sometimes not sinusoidally-varying, and therefore
phasor analysis seems not to apply generally. It is
certainly true that many signals of practical interest
are not sinusoidal, and many are far from it.
Nevertheless, phasor analysis is broadly applicable.
There are basically two reasons why this is so:
• Many signals, although not strictly sinusoidal,
are “narrowband” and therefore well-modeled as
sinusoidal. For example, a cellular
telecommunications signal might have a
bandwidth on the order of 10 MHz and a center
frequency of about 2 GHz. This means the
difference in frequency between the band edges
of this signal is just 0.5% of the center
frequency. The frequency response associated
with signal propagation or with hardware can
often be assumed to be constant over this range
of frequencies. With some caveats, doing phasor
analysis at the center frequency and assuming
the results apply equally well over the bandwidth
of interest is often a pretty good approximation.
• It turns out that phasor analysis is easily
extensible to any physical signal, regardless of
bandwidth. This is so because any physical
signal can be decomposed into a linear
combination of sinusoids – this is known as
Fourier analysis. The way to find this linear
combination of sinusoids is by computing the
Fourier series, if the signal is periodic, or the
Fourier Transform, otherwise. Phasor analysis
applies to each frequency independently, and
(invoking superposition) the results can be added
together to obtain the result for the complete
signal. The process of combining results after
phasor analysis results is nothing more than
integration over frequency; i.e.:
Z +∞
A(t; ω)dω
−∞
Using Equation 1.9, this can be rewritten:
Z +∞
Re C(ω)ejωt dω
−∞
We can go one step further using Claim 2:
Z +∞
jωt
Re
C(ω)e dω
−∞
The quantity in the curly braces is simply the
Fourier transform of C(ω). Thus, we see that we
can analyze a signal of arbitrarily-large
bandwidth simply by keeping ω as an
independent variable while we are doing phasor
analysis, and if we ever need the physical signal,
we just take the real part of the Fourier transform
of the phasor. So not only is it possible to
analyze any time-domain signal using phasor
analysis, it is also often far easier than doing the
same analysis on the time-domain signal directly.
1.6. UNITS
Summarizing:
Phasor analysis does not limit us to sinusoidal
waveforms. Phasor analysis is not only applicable to sinusoids and signals that are sufficiently
narrowband, but is also applicable to signals of
arbitrary bandwidth via Fourier analysis.
13
1.6 Units
[m0072]
The term “unit” refers to the measure used to express
a physical quantity. For example, the mean radius of
the Earth is about 6,371,000 meters; in this case the
unit is the meter.
Additional Reading:
• “Phasor” on Wikipedia.
• “Fourier analysis” on Wikipedia.
A number like “6,371,000” becomes a bit
cumbersome to write, so it is common to use a prefix
to modify the unit. For example, the radius of the
Earth is more commonly said to be 6371 kilometers,
where one kilometer is understood to mean
1000 meters. It is common practice to use prefixes,
such as “kilo-,” that yield values in the range 0.001 to
10, 000. A list of standard prefixes appears in
Table 1.5.
Writing out the names of units can also become
tedious. For this reason, it is common to use standard
abbreviations; e.g., “6731 km” as opposed to
“6371 kilometers,” where “k” is the standard
abbreviation for the prefix “kilo” and “m” is the
standard abbreviation for “meter.” A list of
commonly-used base units and their abbreviations are
shown in Table 1.6.
To avoid ambiguity, it is important to always indicate
the units of a quantity; e.g., writing “6371 km” as
opposed to “6371.” Failure to do so is a common
source of error and misunderstandings. An example is
Prefix
exa
peta
tera
giga
mega
kilo
milli
micro
nano
pico
femto
atto
Abbreviation
E
P
T
G
M
k
m
µ
n
p
f
a
Multiply by:
1018
1015
1012
109
106
103
10−3
10−6
10−9
10−12
10−15
10−18
Table 1.5: Prefixes used to modify units.
CHAPTER 1. PRELIMINARY CONCEPTS
14
Unit
ampere
coulomb
farad
henry
hertz
joule
meter
newton
ohm
second
tesla
volt
watt
weber
Abbreviation
A
C
F
H
Hz
J
m
N
Ω
s
T
V
W
Wb
Quantifies:
electric current
electric charge
capacitance
inductance
frequency
energy
distance
force
resistance
time
magnetic flux density
electric potential
power
magnetic flux
Table 1.6: Some units that are commonly used in electromagnetics.
the expression:
l = 3t
where l is length and t is time. It could be that l is in
meters and t is in seconds, in which case “3” really
means “3 m/s”. However, if it is intended that l is in
kilometers and t is in hours, then “3” really means
“3 km/h” and the equation is literally different. To
patch this up, one might write “l = 3t m/s”; however,
note that this does does not resolve the ambiguity we
just identified – i.e., we still don’t know the units of
the constant “3.” Alternatively, one might write
“l = 3t where l is in meters and t is in seconds,”
which is unambiguous but becomes quite awkward
for more complicated expressions. A better solution is
to write instead:
l = (3 m/s) t
or even better:
In this work, we will use SI units exclusively.
Although SI is probably the most popular for
engineering use overall, other systems remain in
common use. For example, the English system, where
the radius of the Earth might alternatively be said to
be about 3959 miles, continues to be used in various
applications and to a lesser or greater extent in
various regions of the world. An alternative system in
common use in physics and material science
applications is the CGS (“centimeter-gram-second”)
system. The CGS system is similar to SI, but with
some significant differences. For example, the base
unit of energy is the CGS system is not the “joule”
but rather the “erg,” and the values of some physical
constants become unitless. Therefore – once again –
it is very important to include units whenever values
are stated.
SI defines seven fundamental units from which all
other units can be derived. These fundamental units
are distance in meters (m), time in seconds (s),
current in amperes (A), mass in kilograms (kg),
temperature in kelvin (K), particle count in moles
(mol), and luminosity in candela (cd). SI units for
electromagnetic quantities such as coulombs (C) for
charge and volts (V) for electric potential are derived
from these fundamental units.
A frequently-overlooked feature of units is their
ability to assist in error-checking mathematical
expressions. For example, the electric field intensity
may be specified in volts per meter (V/m), so an
expression for the electric field intensity that yields
units of V/m is said to be “dimensionally correct” (but
not necessarily correct), whereas an expression that
cannot be reduced to units of V/m cannot be correct.
Additional Reading:
l = at where a = 3 m/s
since this separates this issue of units from the
perhaps more-important fact that l is proportional to t
and the constant of proportionality (a) is known.
The meter is the fundamental unit of length in the
International System of Units, known by its French
acronym “SI” and sometimes informally referred to
as the “metric system.”
• “International System of Units” on Wikipedia.
• “Centimetre-gram-second system of units” on
Wikipedia.
1.7. NOTATION
15
Z
1.7 Notation
[m0005]
The list below describes notation used in this book.
• Vectors: Boldface is used to indicate a vector;
e.g., the electric field intensity vector will
typically appear as E. Quantities not in boldface
are scalars. When writing by hand, it is common
→
−
to write “E” or “ E ” in lieu of “E.”
• Unit vectors: A circumflex is used to indicate a
unit vector; i.e., a vector having magnitude equal
to one. For example, the unit vector pointing in
the +x direction will be indicated as x̂. In
discussion, the quantity “x̂” is typically spoken
“x hat.”
• Time: The symbol t is used to indicate time.
• Position: The symbols (x, y, z), (ρ, φ, z), and
(r, θ, φ) indicate positions using the Cartesian,
cylindrical, and polar coordinate systems,
respectively. It is sometimes convenient to
express position in a manner which is
independent of a coordinate system; in this case,
we typically use the symbol r. For example,
r = x̂x + ŷy + ẑz in the Cartesian coordinate
system.
• Phasors: A tilde is used to indicate a phasor
quantity; e.g., a voltage phasor might be
indicated as Ve , and the phasor representation of
e
E will be indicated as E.
• Curves, surfaces, and volumes: These
geometrical entities will usually be indicated in
script; e.g., an open surface might be indicated
as S and the curve bounding this surface might
be indicated as C. Similarly, the volume enclosed
by a closed surface S may be indicated as V.
• Integrations over curves, surfaces, and volumes
will usually be indicated using a single integral
sign with the appropriate subscript. For example:
Z
· · · dl is an integral over the curve C
S
· · · ds
is an integral over the surface S
· · · dv
is an integral over the volume V.
• Integrations over closed curves and surfaces will
be indicated using a circle superimposed on the
integral sign. For example:
I
· · · dl is an integral over the closed curve C
C
I
S
··· ds is an integral over the closed surface S
A “closed curve” is one which forms an
unbroken loop; e.g., a circle. A “closed surface”
is one which encloses a volume with no
openings; e.g., a sphere.
• The symbol “∼
=” means “approximately equal
to.” This symbol is used when equality exists,
but is not being expressed with exact numerical
precision. For example, the ratio of the
circumference of a circle to its diameter is π,
where π ∼
= 3.14.
• The symbol “≈” also indicates “approximately
equal to,” but in this case the two quantities are
unequal even if expressed with exact numerical
precision. For example, ex = 1 + x + x2 /2 + ...
as a infinite series, but ex ≈ 1 + x for x ≪ 1.
Using this approximation e0.1 ≈ 1.1, which is in
good agreement with the actual value
e0.1 ∼
= 1.1052.
• The symbol “∼” indicates “on the order of,”
which is a relatively weak statement of equality
indicating that the indicated quantity is within a
factor of 10 or so the indicated value. For
example, µ ∼ 105 for a class of iron alloys, with
exact values being being larger or smaller by a
factor of 5 or so.
• The symbol “,” means “is defined as” or “is
equal as the result of a definition.”
√
• Complex numbers: j , −1.
• See Appendix C for notation used to identify
commonly-used physical constants.
C
Z
V
[m0026]
16
CHAPTER 1. PRELIMINARY CONCEPTS
Image Credits
Fig. 1.1 c V. Blacus, https://commons.wikimedia.org/wiki/File:Electromagnetic-Spectrum.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Fig. 1.2: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0074 fClap.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 1.3: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0074 fTrumpet.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Chapter 2
Electric and Magnetic Fields
2.1 What is a Field?
2.2 Electric Field Intensity
[m0001]
[m0002]
A field is the continuum of values of a quantity
as a function of position and time.
The quantity that the field describes may be a scalar
or a vector, and the scalar part may be either real- or
complex-valued.
In electromagnetics, the electric field intensity E is a
real-valued vector field that may vary as a function of
position and time, and so might be indicated as
“E(x, y, z, t),” “E(r, t),” or simply “E.” When
expressed as a phasor, this quantity is complex-valued
but exhibits no time dependence, so we might say
e
e
instead “E(r)”
or simply “E.”
An example of a scalar field in electromagnetics is the
electric potential, V ; i.e., V (r, t).
A wave is a time-varying field that continues to exist
in the absence of the source that created it and is
therefore able to transport energy.
Electric field intensity is a vector field we assign the
symbol E and has units of electrical potential per
distance; in SI units, volts per meter (V/m). Before
offering a formal definition, it is useful to consider the
broader concept of the electric field.
Imagine that the universe is empty except for a single
particle of positive charge. Next, imagine that a
second positively-charged particle appears; the
situation is now as shown in Figure 2.1. Since like
charges repel, the second particle will be repelled by
the first particle and vice versa. Specifically, the first
particle is exerting force on the second particle. If the
second particle is free to move, it will do so; this is an
expression of kinetic energy. If the second particle is
somehow held in place, we say the second particle
possesses an equal amount of potential energy. This
potential energy is no less “real,” since we can convert
it to kinetic energy simply by releasing the particle,
thereby allowing it to move.
Now let us revisit the original one particle scenario.
F
c M. Goldammer CC BY SA 4.0
Figure 2.1: A positively-charged particle experiences
a repulsive force F in the presence of another particle
which is also positively-charged.
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS
18
(C) respectively, so E has units of N/C. However, we
typically express E in units of V/m, not N/C. What’s
going on? The short answer is that 1 V/m = 1 N/C:
F
N
N·m
J
V
=
=
=
C
C·m
C·m
m
where we have used the fact that 1 N·m = 1 joule (J)
of energy and 1 J/C = 1 V.
c M. Goldammer CC BY SA 4.0
Electric field intensity (E, N/C or V/m) is a vector field that quantifies the force experienced by
a charged particle due to the influence of charge
not associated with that particle.
Figure 2.2: A map of the force that would be experienced by a second particle having a positive charge.
Here, the magnitude and direction of the force is indicated by the size and direction of the arrow.
The analysis of units doesn’t do much to answer the
question of why we should prefer to express E in V/m
as opposed to N/C. Let us now tackle that question.
Note that it is required for the charge to become
vanishingly small (as indicated by taking the limit) in
order for this definition to work. This is because the
source of the electric field is charge, so the test
particle contributes to the total electric field. To
accurately measure the field of interest, the test charge
must be small enough not to significantly perturb the
field. This makes Equation 2.1 awkward from an
engineering perspective, and we’ll address that later
in this section.
Figure 2.3 shows a simple thought experiment that
demonstrates the concept of electric field intensity in
terms of an electric circuit. This circuit consists of a
parallel-plate capacitor in series with a 9 V battery.1
The effect of the battery, connected as shown, is to
force an accumulation of positive charge on the upper
plate, and an accumulation of negative charge on the
lower plate. If we consider the path from the position
labeled “A,” along the wire and through the battery to
the position labeled “B,” the change in electric
potential is +9 V. It must also be true that the change
in electric potential as we travel from B to A through
the capacitor is −9 V, since the sum of voltages over
any closed loop in a circuit is zero. Said differently,
the change in electric potential between the plates of
the capacitor, starting from node A and ending at
node B, is +9 V. Now, note that the spacing between
the plates in the capacitor is 1 mm. Thus, the rate of
change of the potential between the plates is 9 V
divided by 1 mm, which is 9000 V/m. This is literally
the electric field intensity between the plates. That is,
if one places a particle with an infinitesimally-small
charge between the plates (point “C”), and then
measures the ratio of force to charge, one finds it is
9000 N/C pointing toward A. We come to the
following remarkable conclusion:
According the definition of Equation 2.1, the units of
E are those of force divided by charge. The SI units
for force and charge are the newton (N) and coulomb
1 It is not necessary to know anything about capacitors to get to
the point, so no worries!
In that scenario, we could make a map in which every
position in space is assigned a vector that describes
the force that a particle having a specified charge q
would experience if it were to appear there. The result
looks something like Figure 2.2. This map of force
vectors is essentially a description of the electric field
associated with the first particle.
There are many ways in which the electric field may
be quantified. Electric field intensity E is simply one
of these ways. We define E(r) to be the force F(r)
experienced by a test particle having charge q, divided
by q; i.e.,
F(r)
(2.1)
E(r) , lim
q→0 q
2.2. ELECTRIC FIELD INTENSITY
19
The Origin of The Standard Model,” on
YouTube.
B
9V +
1mm
-
A
c Y. Qin CC BY 3.0
Figure 2.3: A simple circuit used to describe the concept of electric field intensity. In this example, E at
point C is 9000 V/m directed from B toward A.
E points in the direction in which electric potential is most rapidly decreasing, and the magnitude
of E is the rate of change in electric potential with
distance in this direction.
The reader may have noticed that we have defined the
electric field in terms of what it does. We have have
not directly addressed the question of what the
electric field is. This is the best we can do using
classical physics, and fortunately, this is completely
adequate for the most engineering applications.
However, a deeper understanding is possible using
quantum mechanics, where we find that the electric
field and the magnetic field are in fact manifestations
of the same fundamental force, aptly named the
electromagnetic force. (In fact, the electromagnetic
force is found to be one of just four fundamental
forces, the others being gravity, the strong nuclear
force, and the weak nuclear force.) Quantum
mechanics also facilitates greater insight into the
nature of electric charge and of the photon, which is
the fundamental constituent of electromagnetic
waves. For more information on this topic, an
excellent starting point is the video “Quantum
Invariance & The Origin of The Standard Model”
referenced at the end of this section.
Additional Reading:
• “Electric field” on Wikipedia.
• PBS Space Time video “Quantum Invariance &
CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS
20
2.3 Permittivity
[m0008]
Permittivity describes the effect of material in
determining the electric field in response to electric
charge. For example, one can observe from laboratory
experiments that a particle having charge q gives rise
to the electric field
E = R̂ q
1 1
4πR2 ǫ
(2.2)
where R is distance from the charge, R̂ is a unit
vector pointing away from the charge, and ǫ is a
constant that depends on the material. Note that E
increases with q, which makes sense since electric
charge is the source of E. Also note that E is
inversely proportional to 4πR2 , indicating that E
decreases in proportion to the area of a sphere
surrounding the charge – a principle commonly
known as the inverse square law. The remaining
factor 1/ǫ is the constant of proportionality, which
captures the effect of material. Given units of V/m for
E and C for Q, we find that ǫ must have units of
farads per meter (F/m). (To see this, note that 1 F =
1 C/V.)
Permittivity (ǫ, F/m) describes the effect of material in determining the electric field intensity in
response to charge.
In free space (that is, a perfect vacuum), we find that
ǫ = ǫ0 where:
ǫ0 ∼
= 8.854 × 10−12 F/m
(2.3)
The permittivity of air is only slightly greater, and
usually can be assumed to be equal to that of free
space. In most other materials, the permittivity is
significantly greater; that is, the same charge results
in a weaker electric field intensity.
It is common practice to describe the permittivity of
materials relative to the permittivity of free space.
This relative permittivity is given by:
ǫr ,
ǫ
ǫ0
(2.4)
Values of ǫr for a few representative materials is
given in Appendix A.1. Note that ǫr ranges from 1
(corresponding to a perfect vacuum) to about 60 or so
in common engineering applications. Also note that
relative permittivity is sometimes referred to as
dielectric constant. This term is a bit misleading,
however, since permittivity is a meaningful concept
for many materials that are not dielectrics.
Additional Reading:
• “Permittivity” on Wikipedia.
• Appendix A.1 (“Permittivity of Some Common
Materials”).
• “Inverse square law” on Wikipedia.
2.4. ELECTRIC FLUX DENSITY
2.4 Electric Flux Density
[m0011]
Electric flux density, assigned the symbol D, is an
alternative to electric field intensity (E) as a way to
quantify an electric field. This alternative description
offers some actionable insight, as we shall point out at
the end of this section.
First, what is electric flux density? Recall that a
particle having charge q gives rise to the electric field
intensity
1 1
E = R̂ q
(2.5)
4πR2 ǫ
where R is distance from the charge and R̂ points
away from the charge. Note that E is inversely
proportional to 4πR2 , indicating that E decreases in
proportion to the area of a sphere surrounding the
charge. Now integrate both sides of Equation 2.5 over
a sphere S of radius R:
I
I 1 1
· ds
(2.6)
E(r) · ds =
R̂ q
4πR2 ǫ
S
S
Factoring out constants that do not vary with the
variables of integration, the right-hand side becomes:
I
1 1
q
R̂ · ds
4πR2 ǫ S
Note that ds = R̂ds in this case, and also that
R̂ · R̂ = 1. Thus, the right-hand side simplifies to:
I
1 1
ds
q
4πR2 ǫ S
The remaining integral is simply the area of S, which
is 4πR2 . Thus, we find:
I
q
E(r) · ds =
(2.7)
ǫ
S
The integral of a vector field over a specified surface
is known as flux (see “Additional Reading” at the end
of this section). Thus, we have found that the flux of
E through the sphere S is equal to a constant, namely
q/ǫ. Furthermore, this constant is the same regardless
of the radius R of the sphere. Said differently, the flux
of E is constant with distance, and does not vary as E
itself does. Let us capitalize on this observation by
21
making the following small modification to
Equation 2.7:
I
S
[ǫE] · ds = q
(2.8)
In other words, the flux of the quantity ǫE is equal to
the enclosed charge, regardless of the radius of the
sphere over which we are doing the calculation. This
leads to the following definition:
The electric flux density D = ǫE, having units
of C/m2 , is a description of the electric field in
terms of flux, as opposed to force or change in
electric potential.
It may appear that D is redundant information given
E and ǫ, but this is true only in homogeneous media.
The concept of electric flux density becomes
important – and decidedly not redundant – when we
encounter boundaries between media having different
permittivities. As we shall see in Section 5.18,
boundary conditions on D constrain the component
of the electric field that is perpendicular to the
boundary separating two regions. If one ignores the
“flux” character of the electric field represented by D
and instead considers only E, then only the tangential
component of the electric field is constrained. In fact,
when one of the two materials comprising the
boundary between two material regions is a perfect
conductor, then the electric field is completely
determined by the boundary condition on D. This
greatly simplifies the problem of finding the electric
field in a region bounded or partially bounded by
materials that can be modeled as perfect conductors,
including many metals. In particular, this principle
makes it easy to analyze capacitors.
We conclude this section with a warning. Even
though the SI units for D are C/m2 , D describes an
electric field and not a surface charge density. It is
certainly true that one may describe the amount of
charge distributed over a surface using units of C/m2 .
However, D is not necessarily a description of actual
charge, and there is no implication that the source of
the electric field is a distribution of surface charge.
On the other hand, it is true that D can be interpreted
as an equivalent surface charge density that would
give rise to the observed electric field, and in some
cases, this equivalent charge density turns out to be
the actual charge density.
CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS
22
Additional Reading:
• “Flux” on Wikipedia.
2.5 Magnetic Flux Density
[m0003]
Magnetic flux density is a vector field which we
identify using the symbol B and which has SI units of
tesla (T). Before offering a formal definition, it is
useful to consider the broader concept of the magnetic
field.
Magnetic fields are an intrinsic property of some
materials, most notably permanent magnets. The
basic phenomenon is probably familiar, and is shown
in Figure 2.4. A bar magnet has “poles” identified as
“N” (“north”) and “S” (“south”). The N-end of one
magnet attracts the S-end of another magnet but
repels the N-end of the other magnet and so on. The
existence of a vector field is apparent since the
observed force acts at a distance and is asserted in a
specific direction. In the case of a permanent magnet,
the magnetic field arises from mechanisms occurring
at the scale of the atoms and electrons comprising the
material. These mechanisms require some additional
explanation which we shall defer for now.
Magnetic fields also appear in the presence of current.
For example, a coil of wire bearing a current is found
to influence permanent magnets (and vice versa) in
the same way that permanent magnets affect each
other. This is shown in Figure 2.5. From this, we infer
that the underlying mechanism is the same – i.e., the
vector field generated by a current-bearing coil is the
same phenomenon as the vector field associated with
a permanent magnet. Whatever the source, we are
S
N
c Y. Qin CC BY 3.0
Figure 2.4: Evidence of a vector field from observations of the force perceived by the bar magnets on the
right in the presence of the bar magnets on the left.
2.5. MAGNETIC FLUX DENSITY
23
B
F
=
B
F
v>
c Y. Qin CC BY 4.0
c Y. Qin CC BY 4.0
Figure 2.5: Evidence that current can also create a
magnetic field.
now interested in quantifying its behavior.
To begin, let us consider the effect of a magnetic field
on a electrically-charged particle. First, imagine a
region of free space with no electric or magnetic
fields. Next, imagine that a charged particle appears.
This particle will experience no force. Next, a
magnetic field appears; perhaps this is due to a
permanent magnet or a current in the vicinity. This
situation is shown in Figure 2.6 (top). Still, no force is
applied to the particle. In fact, nothing happens until
the particle in set in motion. Figure 2.6 (bottom)
shows an example. Suddenly, the particle perceives a
force. We’ll get to the details about direction and
magnitude in a moment, but the main idea is now
evident. A magnetic field is something that applies a
force to a charged particle in motion, distinct from (in
fact, in addition to) the force associated with an
electric field.
Now, it is worth noting that a single charged particle
in motion is the simplest form of current. Remember
also that motion is required for the magnetic field to
influence the particle. Therefore, not only is current
the source of the magnetic field, the magnetic field
also exerts a force on current. Summarizing:
Figure 2.6: The force perceived by a charged particle
that is (top) motionless and (bottom) moving with velocity v = ẑv, which is perpendicular to the plane of
this document and toward the reader.
The magnetic field describes the force exerted on
permanent magnets and currents in the presence
of other permanent magnets and currents.
So, how can we quantify a magnetic field? The
answer from classical physics involves another
experimentally-derived equation that predicts force as
a function of charge, velocity, and a vector field B
representing the magnetic field. Here it is: The force
applied to a particle bearing charge q is
F = qv × B
(2.9)
where v is the velocity of the particle and “×”
denotes the cross product. The cross product of two
vectors is in a direction perpendicular to each of the
two vectors, so the force exerted by the magnetic field
is perpendicular to both the direction of motion and
the direction in which the magnetic field points.
The reader would be well-justified in wondering why
the force exerted by the magnetic field should
perpendicular to B. For that matter, why should the
force depend on v? These are questions for which
classical physics provides no obvious answers.
Effective answers to these questions require concepts
from quantum mechanics, where we find that the
magnetic field is a manifestation of the fundamental
CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS
24
and aptly-named electromagnetic force. The
electromagnetic force also gives rises to the electric
field, and it is only limited intuition, grounded in
classical physics, that leads us to perceive the electric
and magnetic fields as distinct phenomena. For our
present purposes – and for most
commonly-encountered engineering applications –
we do not require these concepts. It is sufficient to
accept this apparent strangeness as fact and proceed
accordingly.
Dimensional analysis of Equation 2.9 reveals that B
has units of (N·s)/(C·m). In SI, this combination of
units is known as the tesla (T).
We refer to B as magnetic flux density, and therefore
tesla is a unit of magnetic flux density. A fair question
to ask at this point is: What makes this a flux density?
The short answer is that this terminology is somewhat
arbitrary, and in fact is not even uniformly accepted.
In engineering electromagnetics, the preference for
referring to B as a “flux density” is because we
frequently find ourselves integrating B over a
mathematical surface. Any quantity that is obtained
by integration over a surface is referred to as “flux,”
and so it becomes natural to think of B as a flux
density; i.e., as flux per unit area. The SI unit for
magnetic flux is the weber (Wb). Therefore, B may
alternatively be described as having units of Wb/m2 ,
and 1 Wb/m2 = 1 T.
c Y. Qin CC BY 4.0
Figure 2.7: The magnetic field of a bar magnet, illustrating field lines.
Magnetic flux density (B, T or Wb/m2 ) is a description of the magnetic field that can be defined
as the solution to Equation 2.9.
When describing magnetic fields, we occasionally
refer to the concept of a field line, defined as follows:
A magnetic field line is the curve in space traced
out by following the direction in which the magnetic field vector points.
This concept is illustrated in Figure 2.7 for a
permanent bar magnet and Figure 2.8 for a
current-bearing coil.
Magnetic field lines are remarkable for the following
reason:
A magnetic field line always forms a closed loop.
c Y. Qin CC BY 4.0
Figure 2.8: The magnetic field of a current-bearing
coil, illustrating field lines.
2.6. PERMEABILITY
This is true in a sense even for field lines which seem
to form straight lines (for example, those along the
axis of the bar magnet and the coil in Figures 2.7 and
2.8), since a field line that travels to infinity in one
direction reemerges from infinity in the opposite
direction.
Additional Reading:
• “Magnetic Field” on Wikipedia.
25
2.6 Permeability
[m0009]
Permeability describes the effect of material in
determining the magnetic flux density. All else being
equal, magnetic flux density increases in proportion to
permeability.
To illustrate the concept, consider that a particle
bearing charge q moving at velocity v gives rise to a
magnetic flux density:
B(r) = µ
qv
× R̂
4πR2
(2.10)
where R̂ is the unit vector pointing from the charged
particle to the field point r, R is this distance, and
“×” is the cross product. Note that B increases with
charge and speed, which makes sense since moving
charge is the source of the magnetic field. Also note
that B is inversely proportional to 4πR2 , indicating
that |B| decreases in proportion to the area of a
sphere surrounding the charge, also known as the
inverse square law. The remaining factor, µ, is the
constant of proportionality that captures the effect of
material. We refer to µ as the permeability of the
material. Since B can be expressed in units of Wb/m2
and the units of v are m/s, we see that µ must have
units of henries per meter (H/m). (To see this, note
that 1 H , 1 Wb/A.)
Permeability (µ, H/m) describes the effect of material in determining the magnetic flux density.
In free space, we find that the permeability µ = µ0
where:
µ0 = 4π × 10−7 H/m
(2.11)
It is common practice to describe the permeability of
materials in terms of their relative permeability:
µr ,
µ
µ0
(2.12)
which gives the permeability relative to the minimum
possible value; i.e., that of free space. Relative
permeability for a few representative materials is
given in Appendix A.2.
Note that µr is approximately 1 for all but a small
class of materials. These are known as magnetic
CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS
26
materials, and may exhibit values of µr as large as
∼ 106 . A commonly-encountered category of
magnetic materials is ferromagnetic material, of
which the best-known example is iron.
Additional Reading:
• “Permeability (electromagnetism)” on
Wikipedia.
• Section 7.16 (“Magnetic Materials”).
2.7 Magnetic Field Intensity
[m0012]
Magnetic field intensity H is an alternative
description of the magnetic field in which the effect of
material is factored out. For example, the magnetic
flux density B (reminder: Section 2.5) due to a point
charge q moving at velocity v can be written in terms
of the Biot-Savart Law:
B=µ
• Appendix A.2 (“Permeability of Some Common
Materials”).
qv
× R̂
4πR2
(2.13)
where R̂ is the unit vector pointing from the charged
particle to the field point r, R is this distance, “×” is
the cross product, and µ is the permeability of the
material. We can rewrite Equation 2.13 as:
B , µH
(2.14)
with:
qv
× R̂
(2.15)
4πR2
so H in homogeneous media does not depend on µ.
H=
Dimensional analysis of Equation 2.15 reveals that
the units for H are amperes per meter (A/m).
However, H does not represent surface current
density,2 as the units might suggest. While it is
certainly true that a distribution of current (A) over
some linear cross-section (m) can be described as a
current density having units of A/m, H is associated
with the magnetic field and not a particular current
distribution. Said differently, H can be viewed as a
description of the magnetic field in terms of an
equivalent (but not actual) current.
The magnetic field intensity H (A/m), defined using Equation 2.14, is a description of the magnetic field independent from material properties.
It may appear that H is redundant information given
B and µ, but this is true only in homogeneous media.
The concept of magnetic field intensity becomes
important – and decidedly not redundant – when we
encounter boundaries between media having different
permeabilities. As we shall see in Section 7.11,
2 The concept of current density is not essential to understand this
section; however, a primer can be found in Section 6.2.
2.8. ELECTROMAGNETIC PROPERTIES OF MATERIALS
boundary conditions on H constrain the component
of the magnetic field which is tangent to the boundary
separating two otherwise-homogeneous regions. If
one ignores the characteristics of the magnetic field
represented by H and instead considers only B, then
only the perpendicular component of the magnetic
field is constrained.
The concept of magnetic field intensity also turns out
to be useful in a certain problems in which µ is not a
constant, but rather is a function of magnetic field
strength. In this case, the magnetic behavior of the
material is said to be nonlinear. For more on this, see
Section 7.16.
Additional Reading:
• “Magnetic field” on Wikipedia.
• “Biot-Savart law” on Wikipedia.
27
2.8 Electromagnetic Properties of
Materials
[m0007]
In electromagnetic analysis, one is principally
concerned with three properties of matter. These
properties are quantified in terms of constitutive
parameters, which describe the effect of material in
determining an electromagnetic quantity in response
to a source. Here are the three principal constitutive
parameters:
• Permittivity (ǫ, F/m) quantifies the effect of
matter in determining the electric field in
response to electric charge. Permittivity is
addressed in Section 2.3.
• Permeability (µ, H/m) quantifies the effect of
matter in determining the magnetic field in
response to current. Permeability is addressed in
Section 2.6.
• Conductivity (σ, S/m) quantifies the effect of
matter in determining the flow of current in
response to an electric field. Conductivity is
addressed in Section 6.3.
The electromagnetic properties of most common
materials in most common applications can be
quantified in terms of the constitutive parameters
ǫ, µ, and σ.
To keep electromagnetic theory from becoming too
complex, we usually require the constitutive
parameters to exhibit a few basic properties. These
properties are as follows:
• Homogeneity. A material that is homogeneous is
uniform over the space it occupies; that is, the
values of its constitutive parameters are constant
at all locations within the material. A
counter-example would be a material that is
composed of multiple chemically-distinct
compounds that are not thoroughly mixed, such
as soil.
CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS
28
• Isotropy. A material that is isotropic behaves in
precisely the same way regardless of how it is
oriented with respect to sources and fields
occupying the same space. A counter-example is
quartz, whose atoms are arranged in a
uniformly-spaced crystalline lattice. As a result,
the electromagnetic properties of quartz can be
changed simply by rotating the material with
respect to the applied sources and fields.
No practical material is truly homogeneous, isotropic,
linear, and time-invariant. However, for most
materials in most applications, the deviation from this
ideal condition is not large enough to significantly
affect engineering analysis and design. In other cases,
materials may be significantly non-ideal in one of
these respects, but may still be analyzed with
appropriate modifications to the theory.
[m0054]
• Linearity. A material is said to be linear if its
properties are constant and independent of the
magnitude of the sources and fields applied to
the material. For example, capacitors have
capacitance, which is determined in part by the
permittivity of the material separating the
terminals (Section 5.23). This material is
approximately linear when the applied voltage V
is below the rated working voltage; i.e., ǫ is
constant and so capacitance does not vary
significantly with respect to V . When V is
greater than the working voltage, the dependence
of ǫ on V becomes more pronounced, and then
capacitance becomes a function of V . In another
practical example, it turns out that µ for
ferromagnetic materials is nonlinear such that
the precise value of µ depends on the magnitude
of the magnetic field.
• Time-invariance. An example of a class of
materials that is not necessarily time-invariant is
piezoelectric materials, for which
electromagnetic properties vary significantly
depending on the mechanical forces applied to
them – a property which can be exploited to
make sensors and transducers.
Linearity and time-invariance (LTI) are particularly
important properties to consider because they are
requirements for superposition. For example, in a LTI
material, we may calculate the field E1 due to a point
charge q1 at r1 and calculate the field E2 due to a
point charge q2 at r2 . Then, when both charges are
simultaneously present, the field is E1 + E2 . The
same is not necessarily true for materials that are not
LTI. Devices that are nonlinear, and therefore not LTI,
do not necessarily follow the rules of elementary
circuit theory, which presume that superposition
applies. This condition makes analysis and design
much more difficult.
2.8. ELECTROMAGNETIC PROPERTIES OF MATERIALS
Image Credits
Fig. 2.1: c M. Goldammer, https://commons.wikimedia.org/wiki/File:M0002 fTwoChargedParticles.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 2.2: c M. Goldammer, https://commons.wikimedia.org/wiki/File:M0002 fForceMap.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 2.3: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0002 fBatCap.svg,
CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/).
Fig. 2.4: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fBarMagnet.svg,
CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/).
Fig. 2.5: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fCoilBarMagnet.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 2.6: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fFqvB.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 2.7: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fFieldLinesBarMagnet.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 2.8: c Y. Qin, https://commons.wikimedia.org/wiki/File:0003 fFieldLinesCoil.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
29
Chapter 3
Transmission Lines
3.1 Introduction to Transmission
Lines
[m0028]
A transmission line is a structure intended to
transport electromagnetic signals or power.
A rudimentary transmission line is simply a pair of
wires with one wire serving as a datum (i.e., a
reference; e.g., “ground”) and the other wire bearing
an electrical potential that is defined relative to that
datum. Transmission lines having random geometry,
such as the test leads shown in Figure 3.1, are useful
only at very low frequencies and when loss,
reactance, and immunity to electromagnetic
interference (EMI) are not a concern.
by Dmitry G
Figure 3.1: These leads used to connect test equipment to circuits in a laboratory are a very rudimentary
form of transmission line, suitable only for very low
frequencies.
However, many circuits and systems operate at
frequencies where the length or cross-sectional
dimensions of the transmission line may be a
significant fraction of a wavelength. In this case, the
transmission line is no longer “transparent” to the
circuits at either end. Furthermore, loss, reactance,
and EMI are significant problems in many
applications. These concerns motivate the use of
particular types of transmission lines, and make it
necessary to understand how to properly connect the
transmission line to the rest of the system.
In electromagnetics, the term “transmission line”
refers to a structure which is intended to support a
guided wave. A guided wave is an electromagnetic
wave that is contained within or bound to the line, and
which does not radiate away from the line. This
condition is normally met if the length and
cross-sectional dimensions of the transmission line
are small relative to a wavelength – say λ/100 (i.e.,
1% of the wavelength). For example, two
randomly-arranged wires might serve well enough to
carry a signal at f = 10 MHz over a length l = 3 cm,
since l is only 0.1% of the wavelength
λ = c/f = 30 m. However, if l is increased to 3 m,
or if f is increased to 1 GHz, then l is now 10% of the
wavelength. In this case, one should consider using a
transmission line that forms a proper guided wave.
Preventing unintended radiation is not the only
concern. Once we have established a guided wave on
a transmission line, it is important that power applied
to the transmission line be delivered to the circuit or
device at the other end and not reflected back into the
source. For the random wire f = 10 MHz, l = 3 cm
example above, there is little need for concern, since
we expect a phase shift of roughly
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
3.2. TYPES OF TRANSMISSION LINES
0.001 · 360◦ = 0.36◦ over the length of the
transmission line, which is about 0.72◦ for a round
trip. So, to a good approximation, the entire
transmission line is at the same electrical potential
and thus transparent to the source and destination.
However, if l is increased to 3 m, or if f is increased
to 1 GHz, then the associated round-trip phase shift
becomes 72◦ . In this case, a reflected signal traveling
in the opposite direction will add to create a total
electrical potential, which varies in both magnitude
and phase with position along the line. Thus, the
impedance looking toward the destination via the
transmission line will be different than the impedance
looking toward the destination directly. (Section 3.15
gives the details.) The modified impedance will
depend on the cross-sectional geometry, materials,
and length of the line.
Cross-sectional geometry and materials also
determine the loss and EMI immunity of the
transmission line.
Summarizing:
Transmission lines are designed to support
guided waves with controlled impedance, low
loss, and a degree of immunity from EMI.
31
3.2 Types of Transmission Lines
[m0144]
Two common types of transmission line are coaxial
line (Figure 3.2) and microstrip line (Figure 3.3).
Both are examples of transverse electromagnetic
(TEM) transmission lines. A TEM line employs a
single electromagnetic wave “mode” having electric
and magnetic field vectors in directions perpendicular
to the axis of the line, as shown in Figures 3.4 and
3.5. TEM transmission lines appear primarily in radio
frequency applications.
TEM transmission lines such as coaxial lines and
microstrip lines are designed to support a single
electromagnetic wave that propagates along the
length of the transmission line with electric and
magnetic field vectors perpendicular to the direction of propagation.
Not all transmission lines exhibit TEM field structure.
In non-TEM transmission lines, the electric and
magnetic field vectors that are not necessarily
perpendicular to the axis of the line, and the structure
of the fields is complex relative to the field structure
of TEM lines. An example of a transmission line that
exhibits non-TEM field structure is the waveguide
(see example in Figure 3.6). Waveguides are most
prevalent at radio frequencies, and tend to appear in
applications where it is important to achieve very low
loss or where power levels are very high. Another
example is common “multimode” optical fiber
(Figure 3.7). Optical fiber exhibits complex field
c Tkgd2007 CC BY 3.0 (modified)
Figure 3.2: Structure of a coaxial transmission line.
CHAPTER 3. TRANSMISSION LINES
32
dielectric slab
metallic trace
gound plane
c SpinningSpark CC BY SA 3.0 (modified)
Figure 3.3:
line.
Structure of a microstrip transmission
c Averse CC BY SA 2.0 Germany
Figure 3.6: A network of radio frequency waveguides
in an air traffic control radar.
structure because the wavelength of light is very small
compared to the cross-section of the fiber, making the
Figure 3.4: Structure of the electric and magnetic excitation and propagation of non-TEM waves
fields within coaxial line. In this case, the wave is difficult to avoid. (This issue is overcome in a
different type of optical fiber, known as “single
propagating away from the viewer.
mode” fiber, which is much more difficult and
expensive to manufacture.)
Higher-order transmission lines, including radiofrequency waveguides and multimode optical
fiber, are designed to guide waves that have relatively complex structure.
Additional Reading:
Figure 3.5: Structure of the electric and magnetic
fields within microstrip line. (The fields outside the
line are possibly significant, complicated, and not
shown.) In this case, the wave is propagating away
from the viewer.
• “Coaxial cable” on Wikipedia.
• “Microstrip” on Wikipedia.
• “Waveguide (electromagnetism)” on Wikipedia.
• “Optical fiber” on Wikipedia.
3.3. TRANSMISSION LINES AS TWO-PORT DEVICES
33
This parameter quantifies the effect of loss in the
line. (See Section 3.8 for details.)
• Characteristic impedance Z0 , having units of Ω.
This is the ratio of potential (“voltage”) to
current when the line is perfectly
impedance-matched at both ends. (See
Section 3.7 for details.)
These parameters depend on the materials and
geometry of the line.
c BigRiz CC BY SA 3.0 Unported
Figure 3.7: Strands of optical fiber.
3.3 Transmission Lines as
Two-Port Devices
Note that a transmission line is typically not
transparent to the source and load. In particular, the
load impedance may be ZL , but the impedance
presented to the source may or may not be equal to
ZL . (See Section 3.15 for more on this concept.)
Similarly, the source impedance may be ZS , but the
impedance presented to the load may or may not be
equal to ZS . The effect of the transmission line on the
source and load impedances will depend on the
parameters identified above.
[m0077]
Figure 3.8 shows common ways to represent
transmission lines in circuit diagrams. In each case,
the source is represented using a Thévenin equivalent
circuit consisting of a voltage source VS in series with
an impedance ZS .1 In transmission line analysis, the
source may also be referred to as the generator. The
termination on the receiving end of the transmission
line is represented, without loss of generality, as an
impedance ZL . This termination is often referred to
as the load, although in practice it can be any circuit
that exhibits an input impedance of ZL .
The two-port representation of a transmission line is
completely described by its length l along with some
combination of the following parameters:
• Phase propagation constant β, having units of
rad/m. This parameter also represents the
wavelength in the line through the relationship
λ = 2π/β. (See Sections 1.3 and 3.8 for details.)
• Attenuation constant α, having units of 1/m.
1 For a refresher on this concept, see “Additional Reading” at the
end of this section.
l
VS
ZS
VS
ZS
ZL
ZL
l
VS
ZS
ZL
c Omegatron CC BY SA 3.0 Unported (modified)
Figure 3.8: Symbols representing transmission lines:
Top: As a generic two-conductor direct connection.
Middle: As a generic two-port “black box.” Bottom:
As a coaxial cable.
CHAPTER 3. TRANSMISSION LINES
34
Additional Reading:
• “Thévenin’s theorem” on Wikipedia.
3.4 Lumped-Element Model
[m0029]
It is possible to ascertain the relevant behaviors of a
transmission line using elementary circuit theory
applied to a differential-length lumped-element model
of the transmission line. The concept is illustrated in
Figure 3.9, which shows a generic transmission line
aligned with its length along the z axis. The
transmission line is divided into segments having
small but finite length ∆z. Each segment is modeled
as an identical two-port having the equivalent circuit
representation shown in Figure 3.10. The equivalent
circuit consists of 4 components as follows:
• The resistance R′ ∆z represents the
series-combined ohmic resistance of the two
conductors. This should account for both
conductors since the current in the actual
transmission line must flow through both
conductors. The prime notation reminds us that
R′ is resistance per unit length; i.e., Ω/m, and it
is only after multiplying by length that we get a
resistance in Ω.
• The conductance G′ ∆z represents the leakage of
current directly from one conductor to the other.
When G′ ∆z > 0, the resistance between the
conductors is less than infinite, and therefore,
current may flow between the conductors. This
amounts to a loss of power separate from the loss
associated with R′ above. G′ has units of S/m.
Further note that G′ is not equal to 1/R′ as
defined above. G′ and R′ are describing entirely
different physical mechanisms (and in principle
either could be defined as either a resistance or a
conductance).
• The capacitance C ′ ∆z represents the
capacitance of the transmission line structure.
Capacitance is the tendency to store energy in
electric fields and depends on the cross-sectional
geometry and the media separating the
conductors. C ′ has units of F/m.
• The inductance L′ ∆z represents the inductance
of the transmission line structure. Inductance is
the tendency to store energy in magnetic fields,
and (like capacitance) depends on the
3.5. TELEGRAPHER’S EQUATIONS
35
3.5 Telegrapher’s Equations
z
[m0079]
z
z
z
z
Figure 3.9: Interpretation of a transmission line as a
cascade of discrete series-connected two-ports.
R'Δz
L'Δz
G'Δz
C'Δz
c Omegatron CC BY SA 3.0 Unported (modified)
Figure 3.10:
Lumped-element equivalent circuit
model for each of the two-ports in Figure 3.9.
cross-sectional geometry and the media
separating the conductors. L′ has units of H/m.
In order to use the model, one must have values for
R′ , G′ , C ′ , and L′ . Methods for computing these
parameters are addressed elsewhere in this book.
In this section, we derive the equations that govern
the potential v(z, t) and current i(z, t) along a
transmission line that is oriented along the z axis. For
this, we will employ the lumped-element model
developed in Section 3.4.
To begin, we define voltages and currents as shown in
Figure 3.11. We assign the variables v(z, t) and
i(z, t) to represent the potential and current on the left
side of the segment, with reference polarity and
direction as shown in the figure. Similarly we assign
the variables v(z + ∆z, t) and i(z + ∆z, t) to
represent the potential and current on the right side of
the segment, again with reference polarity and
direction as shown in the figure. Applying Kirchoff’s
voltage law from the left port, through R′ ∆z and
L′ ∆z, and returning via the right port, we obtain:
∂
i(z, t)
∂t
− v(z + ∆z, t) = 0 (3.1)
v(z, t) − (R′ ∆z) i(z, t) − (L′ ∆z)
Moving terms referring to current to the right side of
the equation and then dividing through by ∆z, we
obtain
v(z + ∆z, t) − v(z, t)
=
∆z
−
R′ i(z, t) + L′
∂
i(z, t)
∂t
(3.2)
Then taking the limit as ∆z → 0:
−
∂
∂
v(z, t) = R′ i(z, t) + L′ i(z, t)
∂z
∂t
R'Δz
L'Δz
+
+
v(z,t)
_
(3.3)
i(z,t)
G'Δz
i(z+Δz,t) v(z+Δz,t)
C'Δz
_
c Omegatron CC BY SA 3.0 Unported (modified)
Figure 3.11:
Lumped-element equivalent circuit
transmission line model, annotated with sign conventions for potentials and currents.
CHAPTER 3. TRANSMISSION LINES
36
Applying Kirchoff’s current law at the right port, we
obtain:
∂
v(z+∆z, t)
∂t
− i(z + ∆z, t) = 0 (3.4)
i(z, t)−(G′ ∆z) v(z+∆z, t)−(C ′ ∆z)
Moving terms referring to potential to the right side of
the equation and then dividing through by ∆z, we
obtain
−
i(z + ∆z, t) − i(z, t)
=
∆z
G′ v(z + ∆z, t) + C ′
∂
v(z + ∆z, t)
∂t
∂
∂
i(z, t) = G′ v(z, t) + C ′ v(z, t)
∂z
∂t
Now we see:
o
n
o
e ejωt
i(z, t) = Re I(z)
n
o
∂
∂
v(z, t) =
Re Ve (z) ejωt
∂z
∂z ∂ e
= Re
V (z) ejωt
∂z
2 For a refresher on phasor analysis, see Section 1.5.
−
(3.6)
−
The time-domain telegrapher’s equations are usually
more than we need or want. If we are only interested
in the response to a sinusoidal stimulus, then
considerable simplification is possible using phasor
representation.2 First we define phasors Ve (z) and
e through the usual relationship:
I(z)
v(z, t) = Re Ve (z) ejωt
In other words, ∂i(z, t)/∂t expressed in phasor
e
representation is jω I(z).
Therefore, Equation 3.3
expressed in phasor representation is:
∂ e
e
V (z) = [R′ + jωL′ ] I(z)
∂z
(3.9)
Following the same procedure, Equation 3.6
expressed in phasor representation is found to be:
Equations 3.3 and 3.6 are the telegrapher’s equations. These coupled (simultaneous) differential
equations can be solved for v(z, t) and i(z, t)
given R′ , G′ , L′ , C ′ and suitable boundary conditions.
n
n
o
∂
∂
e ejωt
i(z, t) = Re I(z)
∂t
∂t i
∂ he
= Re
I(z)ejωt
∂t
o
nh
i
e
= Re jω I(z)
ejωt
(3.5)
Taking the limit as ∆z → 0:
−
In other words, ∂v(z, t)/∂z expressed in phasor
representation is simply ∂ Ve (z)/∂z; and
(3.7)
(3.8)
∂ e
I(z) = [G′ + jωC ′ ] Ve (z)
∂z
(3.10)
Equations 3.9 and 3.10 are the telegrapher’s
equations in phasor representation.
The principal advantage of these equations over the
time-domain versions is that we no longer need to
contend with derivatives with respect to time – only
derivatives with respect to distance remain. This
considerably simplifies the equations.
Additional Reading:
• “Telegrapher’s equations” on Wikipedia.
• “Kirchhoff’s circuit laws” on Wikipedia.
3.6. WAVE EQUATION FOR A TEM TRANSMISSION LINE
3.6 Wave Equation for a TEM
Transmission Line
[m0027]
Consider a TEM transmission line aligned along the z
axis. The phasor form of the Telegrapher’s Equations
(Section 3.5) relate the potential phasor Ve (z) and the
e to each other and to the
current phasor I(z)
lumped-element model equivalent circuit parameters
R′ , G′ , C ′ , and L′ . These equations are
−
−
∂ e
e
V (z) = [R′ + jωL′ ] I(z)
∂z
∂ e
I(z) = [G′ + jωC ′ ] Ve (z)
∂z
(3.11)
(3.12)
An obstacle to using these equations is that we
require both equations to solve for either the potential
or the current. In this section, we reduce these
equations to a single equation – a wave equation –
that is more convenient to use and provides some
additional physical insight.
We begin by differentiating both sides of
Equation 3.11 with respect to z, yielding:
37
The propagation constant γ (units of m−1 ) captures the effect of materials, geometry, and frequency in determining the variation in potential
and current with distance on a TEM transmission
line.
Following essentially the same procedure but
beginning with Equation 3.12, we obtain
∂2 e
e =0
I(z) − γ 2 I(z)
∂z 2
(3.18)
Equations 3.15 and 3.18 are the wave equations
e
for Ve (z) and I(z),
respectively.
e satisfy the same linear
Note that both Ve (z) and I(z)
homogeneous differential equation. This does not
e are equal. Rather, it means
mean that Ve (z) and I(z)
e
e
that V (z) and I(z) can differ by no more than a
multiplicative constant. Since Ve (z) is potential and
e is current, that constant must be an impedance.
I(z)
This impedance is known as the characteristic
impedance and is determined in Section 3.7.
The general solutions to Equations 3.15 and 3.18 are
∂ e
∂2
I(z)
− 2 Ve (z) = [R′ + jωL′ ]
∂z
∂z
(3.13)
e
Then using Equation 3.12 to eliminate I(z),
we obtain
∂2 e
V (z) = − [R′ + jωL′ ] [G′ + jωC ′ ] Ve (z)
∂z 2
(3.14)
This equation is normally written as follows:
−
∂2 e
V (z) − γ 2 Ve (z) = 0
∂z 2
(3.15)
where we have made the substitution:
γ 2 = (R′ + jωL′ ) (G′ + jωC ′ )
(3.16)
The principal square root of γ 2 is known as the
propagation constant:
p
γ , (R′ + jωL′ ) (G′ + jωC ′ )
(3.17)
Ve (z) = V0+ e−γz + V0− e+γz
e = I + e−γz + I − e+γz
I(z)
0
0
(3.19)
(3.20)
where V0+ , V0− , I0+ , and I0− are complex-valued
constants. It is shown in Section 3.8 that
Equations 3.19 and 3.20 represent sinusoidal waves
propagating in the +z and −z directions along the
length of the line. The constants may represent
sources, loads, or simply discontinuities in the
materials and/or geometry of the line. The values of
the constants are determined by boundary conditions;
e at some position(s)
i.e., constraints on Ve (z) and I(z)
along the line.
The reader is encouraged to verify that the
Equations 3.19 and 3.20 are in fact solutions to
Equations 3.15 and 3.18, respectively, for any values
of the constants V0+ , V0− , I0+ , and I0− .
CHAPTER 3. TRANSMISSION LINES
38
3.7 Characteristic Impedance
e at some
conditions; i.e., constraints on Ve (z) and I(z)
position(s) along the line. Also, we will make use of
the telegrapher’s equations (Section 3.5):
[m0052]
Characteristic impedance is the ratio of voltage to
current for a wave that is propagating in single
direction on a transmission line. This is an important
parameter in the analysis and design of circuits and
systems using transmission lines. In this section, we
formally define this parameter and derive an
expression for this parameter in terms of the
equivalent circuit model introduced in Section 3.4.
Consider a transmission line aligned along the z axis.
Employing some results from Section 3.6, recall that
the phasor form of the wave equation in this case is
∂2 e
V (z) − γ 2 Ve (z) = 0
∂z 2
where
γ,
p
(R′ + jωL′ ) (G′ + jωC ′ )
(3.22)
(3.23)
e satisfy the same linear
Since both Ve (z) and I(z)
homogeneous differential equation, they may differ
by no more than a multiplicative constant. Since Ve (z)
e is current, that constant can be
is potential and I(z)
expressed in units of impedance. Specifically, this is
the characteristic impedance, so-named because it
depends only on the materials and cross-sectional
geometry of the transmission line – i.e., things which
determine γ – and not length, excitation, termination,
or position along the line.
To derive the characteristic impedance, first recall that
the general solutions to Equations 3.21 and 3.23 are
Ve (z) = V0+ e−γz + V0− e+γz
e = I + e−γz + I − e+γz
I(z)
0
0
∂ e
e
V (z) = [R′ + jωL′ ] I(z)
∂z
(3.24)
(3.25)
where V0+ , V0− , I0+ , and I0− are complex-valued
constants whose values are determined by boundary
(3.26)
∂ e
I(z) = [G′ + jωC ′ ] Ve (z)
(3.27)
∂z
We begin by differentiating Equation 3.24 with
respect to z, which yields
−
∂ e
V (z) = −γ V0+ e−γz − V0− e+γz
∂z
(3.28)
Now we use this this to eliminate ∂ Ve (z)/∂z in
Equation 3.26, yielding
e
γ V + e−γz − V − e+γz = [R′ + jωL′ ] I(z)
(3.29)
0
(3.21)
Equation 3.21 relates the potential phasor Ve (z) to the
equivalent circuit parameters R′ , G′ , C ′ , and L′ . An
equation of the same form relates the current phasor
e to the equivalent circuit parameters:
I(z)
∂2 e
e =0
I(z) − γ 2 I(z)
∂z 2
−
0
e yields:
Solving the above equation for I(z)
e =
I(z)
γ
R′ + jωL′
V0+ e−γz − V0− e+γz
Comparing this to Equation 3.25, we note
γ
(3.30)
I0+ =
R′ + jωL′
V0+
(3.31)
I0− =
R′ + jωL′
−γ
V0−
(3.32)
We now make the substitution
R′ + jωL′
γ
(3.33)
−V0−
V0+
=
, Z0
I0+
I0−
(3.34)
Z0 =
and observe
As anticipated, we have found that coefficients in the
equations for potentials and currents are related by an
impedance, namely, Z0 .
Characteristic impedance can be written entirely in
terms of the equivalent circuit parameters by
substituting Equation 3.22 into Equation 3.33,
yielding:
s
R′ + jωL′
Z0 =
(3.35)
G′ + jωC ′
3.8. WAVE PROPAGATION ON A TEM TRANSMISSION LINE
The characteristic impedance Z0 (Ω) is the ratio
of potential to current in a wave traveling in a
single direction along the transmission line.
39
3.8 Wave Propagation on a TEM
Transmission Line
[m0080]
Take care to note that Z0 is not the ratio of Ve (z) to
e in general; rather, Z0 relates only the potential
I(z)
and current waves traveling in the same direction.
Finally, note that transmission lines are normally
designed to have a characteristic impedance that is
completely real-valued – that is, with no imaginary
component. This is because the imaginary component
of an impedance represents energy storage (think of
capacitors and inductors), whereas the purpose of a
transmission line is energy transfer.
Additional Reading:
• “Characteristic impedance” on Wikipedia.
In Section 3.6, it is shown that expressions for the
phasor representations of the potential and current
along a transmission line are
Ve (z) = V0+ e−γz + V0− e+γz
e = I + e−γz + I − e+γz
I(z)
0
0
(3.36)
(3.37)
where γ is the propagation constant and it assumed
that the transmission line is aligned along the z axis.
In this section, we demonstrate that these expressions
represent sinusoidal waves, and point out some
important features. Before attempting this section, the
reader should be familiar with the contents of
Sections 3.4, 3.6, and 3.7. A refresher on fundamental
wave concepts (Section 1.3) may also be helpful.
We first define real-valued quantities α and β to be
the real and imaginary components of γ; i.e.,
α , Re {γ}
(3.38)
β , Im {γ}
(3.39)
γ = α + jβ
(3.40)
and subsequently
Then we observe
e±γz = e±(α+jβ)z = e±αz e±jβz
(3.41)
It may be easier to interpret this expression by
reverting to the time domain:
Re e±γz ejωt = e±αz cos (ωt ± βz)
(3.42)
Thus, e−γz represents a damped sinusoidal wave
traveling in the +z direction, and e+γz represents a
damped sinusoidal wave traveling in the −z direction.
Let’s define Ve + (z) and Ie+ (z) to be the potential and
current associated with a wave propagating in the +z
direction. Then:
Ve + (z) , V0+ e−γz
(3.43)
CHAPTER 3. TRANSMISSION LINES
40
v+(z,t=0)
transmission line is said to be lossless. If α > 0 then
the line is said to be lossy (or possibly “low loss” if
the loss can be neglected), and in this case the rate at
which the magnitude decreases with distance
increases with α.
z Next let us consider the speed of the wave. To answer
Figure 3.12: The potential v + (z, t) of the wave traveling in the +z direction at t = 0 for ψ = 0.
or equivalently in the time domain:
n
o
v + (z, t) = Re Ve + (z) ejωt
= Re V0+ e−γz ejωt
= V0+ e−αz cos (ωt − βz + ψ)
where φ is some constant. Similarly, the phase at
some time ∆t later and some point ∆z further along
can be written as
(3.44)
where ψ is the phase of V0+ . Figure 3.12 shows
v + (z, t). From fundamental wave theory we
recognize
The phase velocity vp is ∆z/∆t when these two
phases are equal; i.e., when
Solving for vp = ∆z/∆t, we obtain:
vp =
ω
β
(3.46)
Having previously noted that β = 2π/λ, the above
expression also yields the expected result
Subsequently the wavelength in the line is
2π
β
ω (t + ∆t) − β (z + ∆z) + φ
ωt − βz + φ = ω (t + ∆t) − β (z + ∆z) + φ
β , Im {γ} (rad/m) is the phase propagation constant, which is the rate at which phase
changes as a function of distance.
λ=
this question, we need to be a bit more specific about
what we mean by “speed.” At the moment, we mean
phase velocity; that is, the speed at which a point of
constant phase seems to move through space. In other
words, what distance ∆z does a point of constant
phase traverse in time ∆t? To answer this question,
we first note that the phase of v + (z, t) can be written
generally as
ωt − βz + φ
(3.45)
Also we recognize:
α , Re {γ} (1/m) is the attenuation constant,
which is the rate at which magnitude diminishes
as a function of distance.
Sometimes the units of α are indicated as “Np/m”
(“nepers” per meter), where the term “neper” is used
to indicate the units of the otherwise unitless
real-valued exponent of the constant e.
Note that α = 0 for a wave that does not diminish in
magnitude with increasing distance, in which case the
vp = λf
(3.47)
The phase velocity vp = ω/β = λf is the speed
at which a point of constant phase travels along
the line.
Returning now to consider the current associated with
the wave traveling in the +z direction:
Ie+ (z) = I0+ e−γz
(3.48)
We can rewrite this expression in terms of the
characteristic impedance Z0 , as follows:
+
V
Ie+ (z) = 0 e−γz
Z0
(3.49)
3.9. LOSSLESS AND LOW-LOSS TRANSMISSION LINES
Similarly, we find that the current Ie− (z) associated
with Ve − (z) for the wave traveling in the −z direction
is
−V0− −γz
Ie− (z) =
e
(3.50)
Z0
The negative sign appearing in the above expression
emerges as a result of the sign conventions used for
potential and current in the derivation of the
telegrapher’s equations (Section 3.5). The physical
significance of this change of sign is that wherever the
potential of the wave traveling in the −z direction is
positive, then the current at the same point is flowing
in the −z direction.
It is frequently necessary to consider the possibility
that waves travel in both directions simultaneously. A
very important case where this arises is when there is
reflection from a discontinuity of some kind; e.g.,
from a termination which is not perfectly
impedance-matched. In this case, the total potential
e can be expressed as the
Ve (z) and total current I(z)
general solution to the wave equation; i.e., as the sum
of the “incident” (+z-traveling) wave and the
reflected (−z-traveling) waves:
Ve (z) = Ve + (z) + Ve − (z)
e = Ie+ (z) + Ie− (z)
I(z)
(3.51)
41
3.9 Lossless and Low-Loss
Transmission Lines
[m0083]
Quite often the loss in a transmission line is small
enough that it may be neglected. In this case, several
aspects of transmission line theory may be simplified.
In this section, we present these simplifications.
First, recall that “loss” refers to the reduction of
magnitude as a wave propagates through space. In the
lumped-element equivalent circuit model
(Section 3.4), the parameters R′ and G′ of the
represent physical mechanisms associated with loss.
Specifically, R′ represents the resistance of
conductors, whereas G′ represents the undesirable
current induced between conductors through the
spacing material. Also recall that the propagation
constant γ is, in general, given by
p
γ , (R′ + jωL′ ) (G′ + jωC ′ )
(3.53)
With this in mind, we now define “low loss” as
meeting the conditions:
(3.52)
R′ ≪ ωL′
(3.54)
The existence of waves propagating simultaneously in
both directions gives rise to a phenomenon known as
a standing wave. Standing waves and the calculation
of the coefficients V0− and I0− due to reflection are
addressed in Sections 3.13 and 3.12 respectively.
G′ ≪ ωC ′
(3.55)
When these conditions are met, the propagation
constant simplifies as follows:
p
γ ≈ (jωL′ ) (jωC ′ )
p
= −ω 2 L′ C ′
√
(3.56)
= jω L′ C ′
and subsequently
α , Re {γ} ≈ 0
(low-loss approx.) (3.57)
√
β , Im {γ} ≈ ω L′ C ′
1
vp = ω/β ≈ √
L′ C ′
Similarly:
s
Z0 =
R′ + jωL′
≈
G′ + jωC ′
(low-loss approx.) (3.58)
(low-loss approx.) (3.59)
r
L′
(low-loss approx.)
C′
(3.60)
CHAPTER 3. TRANSMISSION LINES
42
Of course if the line is strictly lossless (i.e.,
R′ = G′ = 0) then these are not approximations, but
rather the exact expressions.
b
a
In practice, these approximations are quite commonly
used, since practical transmission lines typically meet
the conditions expressed in Inequalities 3.54 and 3.55
and the resulting expressions are much simpler. We
further observe that Z0 and vp are approximately
independent of frequency when these conditions hold.
However, also note that “low loss” does not mean “no
loss,” and it is common to apply these expressions
even when R′ and/or G′ is large enough to yield
significant loss. For example, a coaxial cable used to
connect an antenna on a tower to a radio near the
ground typically has loss that is important to consider
in the analysis and design process, but nevertheless
satisfies Equations 3.54 and 3.55. In this case, the
low-loss expression for β is used, but α might not be
approximated as zero.
σs s
Figure 3.13: Cross-section of a coaxial transmission
line, indicating design parameters.
3.10 Coaxial Line
[m0143]
Coaxial transmission lines consists of metallic inner
and outer conductors separated by a spacer material
as shown in Figure 3.13. The spacer material is
typically a low-loss dielectric material having
permeability approximately equal to that of free space
(µ ≈ µ0 ) and permittivity ǫs that may range from
very near ǫ0 (e.g., air-filled line) to 2–3 times ǫ0 . The
outer conductor is alternatively referred to as the
“shield,” since it typically provides a high degree of
isolation from nearby objects and electromagnetic
fields. Coaxial line is single-ended3 in the sense that
the conductor geometry is asymmetric and the shield
is normally attached to ground at both ends. These
characteristics make coaxial line attractive for
connecting single-ended circuits in widely-separated
locations and for connecting antennas to receivers and
transmitters.
Coaxial lines exhibit TEM field structure as shown in
Figure 3.14.
Expressions for the equivalent circuit parameters C ′
and L′ for coaxial lines can be obtained from basic
electromagnetic theory. It is shown in Section 5.24
that the capacitance per unit length is
C′ =
2πǫs
ln (b/a)
(3.61)
where a and b are the radii of the inner and outer
conductors, respectively. Using analysis shown in
3 The references in “Additional Reading” at the end of this section
may be helpful if you are not familiar with this concept.
3.10. COAXIAL LINE
43
Since the low-loss conditions are often met, a
convenient expression for the characteristic
impedance is obtained from Equations 3.61 and 3.62
for L′ and C ′ respectively:
r
L′
Z0 ≈
(low-loss)
C′
r
b
1
µ0
ln
(3.64)
=
2π ǫs
a
Figure 3.14: Structure of the electric and magnetic
fields within coaxial line. In this case, the wave is
propagating away from the viewer.
Section 7.14, the inductance per unit length is
b
µ0
′
(3.62)
ln
L =
2π
a
The loss conductance G′ depends on the conductance
σs of the spacer material, and is given by
G′ =
2πσs
ln (b/a)
(3.63)
This expression is derived in Section 6.5.
The resistance per unit length, R′ , is relatively
difficult to quantify. One obstacle is that the inner and
outer conductors typically consist of different
materials or compositions of materials. The inner
conductor is not necessarily a single homogeneous
material; instead, the inner conductor may consist of a
variety of materials selected by trading-off between
conductivity, strength, weight, and cost. Similarly, the
outer conductor is not necessarily homogeneous; for a
variety of reasons, the outer conductor may instead be
a metal mesh, a braid, or a composite of materials.
Another complicating factor is that the resistance of
the conductor varies significantly with frequency,
whereas C ′ , L′ , and G′ exhibit relatively little
variation from their electro- and magnetostatic values.
These factors make it difficult to devise a single
expression for R′ that is both as simple as those
shown above for the other parameters and generally
applicable. Fortunately, it turns out that the low-loss
conditions R′ ≪ ωL′ and G′ ≪ ωC ′ are often
applicable,4 so that R′ and G′ are important only if it
is necessary to compute loss.
4 See Section 3.9 for a reminder about this concept.
The spacer permittivity can be expressed as ǫs = ǫr ǫ0
where ǫr is the relative
permittivity of the spacer
p
material. Since µ0 /ǫ0 is a constant, the above
expression is commonly written
60 Ω b
Z0 ≈ √ ln
(low-loss)
ǫr
a
(3.65)
Thus, it is possible to express Z0 directly in terms of
parameters describing the geometry (a and b) and
material (ǫr ) used in the line, without the need to first
compute the values of components in the
lumped-element equivalent circuit model.
Similarly, the low-loss approximation makes it
possible to express the phase velocity νp directly in
terms the spacer permittivity:
1
(low-loss)
L′ C ′
c
=√
ǫr
νp ≈ √
(3.66)
√
since c , 1/ µ0 ǫ0 . In other words, the phase
velocity in a low-loss coaxial line is approximately
equal to the speed of electromagnetic propagation in
free space, divided by the square root of the relative
permittivity of the spacer material. Therefore, the
phase velocity in an air-filled coaxial line is
approximately equal to speed of propagation in free
space, but is reduced in a coaxial line using a
dielectric spacer.
Example 3.1. RG-59 Coaxial Cable.
RG-59 is a very common type of coaxial line.
Figure 3.15 shows a section of RG-59 cut away
so as to reveal its structure. The radii are
a∼
= 0.292 mm and b ≈ 1.855 mm (mean),
yielding L′ ≈ 370 nH/m. The spacer material is
polyethylene having ǫr ∼
= 2.25, yielding
CHAPTER 3. TRANSMISSION LINES
44
C ′ ≈ 67.7 pF/m. The conductivity of
polyethylene is σs ∼
= 5.9 × 10−5 S/m, yielding
′
G ≈ 200 µS/m. Typical resistance per unit
length R′ is on the order of 0.1 Ω/m near DC,
increasing approximately in proportion to the
square root of frequency.
From the above values, we find that RG-59
satisfies the low-loss criteria R′ ≪ ωL′ for
f ≫ 43 kHz and G′ ≪ ωC ′ for f ≫ 470 kHz.
Underp
these conditions, we find
Z0 ≈ L′ /C ′ ∼
= 74 Ω. Thus, the ratio of the
potential to the current in a wave traveling in a
single direction on RG-59 is about 74 Ω.
The phase
is found to be
√ velocity of RG-59
vp ≈ 1/ L′ C ′ ∼
= 2 × 108 m/s, which is about
67% of c. In other words, a signal that takes 1 ns
to traverse a distance l in free space requires
about 1.5 ns to traverse a length-l section of
RG-59. Since vp = λf , a wavelength in RG-59
is 67% of a wavelength in free space.
Using the expression
p
γ = (R′ + jωL′ ) (G′ + jωC ′ )
c Arj CC BY SA 3.0
Figure 3.15: RG-59 coaxial line. A: Insulating jacket.
B: Braided outer conductor. C: Dielectric spacer. D:
Inner conductor.
t
0
r
(3.67)
with R′ = 0.1 Ω/m, and then taking the real part
to obtain α, we find α ∼ 0.01 m−1 . So, for
example, the magnitude of the potential or
current is decreased by about 50% by traveling a
distance of about 70 m. In other words,
e−αl = 0.5 for l ∼ 70 m at relatively low
frequencies, and increases with increasing
frequency.
Additional Reading:
• “Coaxial cable” on Wikipedia. Includes
descriptions and design parameters for a variety
of commonly-encountered coaxial cables.
• “Single-ended signaling” on Wikipedia.
• Sec. 8.7 (“Differential Circuits”) in
S.W. Ellingson, Radio Systems Engineering,
Cambridge Univ. Press, 2016.
l
W
h
c 7head7metal7 CC BY SA 3.0 (modified)
Figure 3.16: Microstrip transmission line structure
and design parameters.
3.11 Microstrip Line
[m0082]
A microstrip transmission line consists of a narrow
metallic trace separated from a metallic ground plane
by a slab of dielectric material, as shown in
Figure 3.16. This is a natural way to implement a
transmission line on a printed circuit board, and so
accounts for an important and expansive range of
applications. The reader should be aware that
microstrip is distinct from stripline, which is a very
different type of transmission line; see “Additional
Reading” at the end of this section for disambiguation
of these terms.
A microstrip line is single-ended5 in the sense that the
5 The reference in “Additional Reading” at the end of this section
may be helpful if you are not familiar with this concept.
3.11. MICROSTRIP LINE
45
W
conductor geometry is asymmetric and the one
conductor – namely, the ground plane – also normally
serves as ground for the source and load.
The spacer material is typically a low-loss dielectric
material having permeability approximately equal to
that of free space (µ ≈ µ0 ) and relative permittivity ǫr
in the range 2 to about 10 or so.
A microstrip line nominally exhibits TEM field
structure. This structure is shown in Figure 3.17.
Note that electric and magnetic fields exist both in the
dielectric and in the space above the dielectric, which
is typically (but not always) air. This complex field
structure makes it difficult to describe microstrip line
concisely in terms of the equivalent circuit parameters
of the lumped-element model. Instead, expressions
for Z0 directly in terms of h/W and ǫr are typically
used instead. A variety of these expressions are in
common use, representing different approximations
and simplifications. A widely-accepted and
broadly-applicable expression is:6
42.4 Ω
Z0 ≈ √
×
ǫr + 1
!#
"
r
1 + 1/ǫr 2
4h
2
π
ln 1 + ′ Φ + Φ +
W
2
(3.68)
where
Φ,
14 + 8/ǫr
11
4h
W′
(3.69)
and W ′ is W adjusted to account for the thickness t
of the microstrip line. Typically t ≪ W and t ≪ h,
for which W ′ ≈ W . Simpler approximations for Z0
are also commonly employed in the design and
analysis of microstrip lines. These expressions are
limited in the range of h/W for which they are valid,
and can usually be shown to be special cases or
approximations of Equation 3.68. Nevertheless, they
are sometimes useful for quick “back of the envelope”
calculations.
h
Figure 3.17: Structure of the electric and magnetic
fields within microstrip line. (The fields outside the
line are possibly significant, complicated, and not
shown.) In this case, the wave is propagating away
from the viewer.
the theory of uniform plane waves in unbounded
media (Equation 9.38 from Section 9.2):
√
β = ω µǫ
(3.70)
It turns out that the electromagnetic field structure in
the space between the conductors is
well-approximated as that of a uniform plane wave in
unbounded media having the same permeability µ0
but a different relative permittivity, which we shall
assign the symbol ǫr,ef f (for “effective relative
permittivity”). Then
√
β ≈ ω µ0 ǫr,ef f ǫ0 (low-loss microstrip)
√
(3.71)
= β0 ǫr,ef f
In other words, the phase propagation constant in a
microstrip line can be approximated as the free-space
√
phase propagation β0 , ω µ0 ǫ0 times a correction
√
factor ǫr,ef f . Then ǫr,ef f may be crudely
approximated as follows:
ǫr,ef f ≈
ǫr + 1
2
(3.72)
i.e., ǫr,ef f is roughly the average of the relative
permittivity of the dielectric slab and the relative
permittivity of free space. The assumption employed
here is that ǫr,ef f is approximately the average of
these values because some fraction of the power in
Accurate expressions for wavelength λ, phase
the guided wave is in the dielectric, and the rest is
propagation constant β, and phase velocity vp are
above the dielectric. Various approximations are
similarly difficult to obtain for waves in microstrip
available to improve on this approximation; however,
line. An approximate technique employs a result from in practice variations in the value of ǫ for the
r
6 This is from Wheeler 1977, cited in “Additional Reading” at the
dielectric due to manufacturing processes typically
make a more precise estimate irrelevant.
end of this section.
CHAPTER 3. TRANSMISSION LINES
46
Using this concept, we obtain
λ=
2π
λ0
2π
= √
=√
β
β0 ǫr,ef f
ǫr,ef f
(3.73)
where λ0 is the free-space wavelength c/f . Similarly
the phase velocity vp , can be estimated using the
relationship
c
ω
(3.74)
vp = = √
β
ǫr,ef f
i.e., the phase velocity in microstrip is slower than c
√
by a factor of ǫr,ef f .
Example 3.2. 50 Ω Microstrip in FR4 Printed
Circuit Boards.
FR4 is a low-loss fiberglass epoxy dielectric that
is commonly used to make printed circuit boards
(see “Additional Reading” at the end of this
section). FR4 circuit board material is
commonly sold in a slab having thickness
h∼
= 4.5. Let us consider
= 1.575 mm with ǫr ∼
how we might implement a microstrip line
having Z0 = 50 Ω using this material. Since h
and ǫr are fixed, the only parameter remaining to
set Z0 is W . A bit of experimentation with
Equation 3.68 reveals that h/W ≈ 1/2 yields
Z0 ≈ 50 Ω for ǫr = 4.5. Thus, W should be
about 3.15 mm. The effective relative
permittivity is
ǫr,ef f ≈ (4.5 + 1)/2 = 2.75
so the phase velocity
for the wave guided by this
√
line is about c/ 2.75; i.e., 60% of c. Similarly,
the wavelength of this wave is about 60% of the
free space wavelength.
Additional Reading:
• “Microstrip” on Wikipedia.
• “Printed circuit board” on Wikipedia.
• “Stripline” on Wikipedia.
• “Single-ended signaling” on Wikipedia.
• Sec. 8.7 (“Differential Circuits”) in
S.W. Ellingson, Radio Systems Engineering,
Cambridge Univ. Press, 2016.
• H.A. Wheeler, “Transmission Line Properties of
a Strip on a Dielectric Sheet on a Plane,” IEEE
Trans. Microwave Theory & Techniques, Vol. 25,
No. 8, Aug 1977, pp. 631–47.
• “FR-4” on Wikipedia.
3.12. VOLTAGE REFLECTION COEFFICIENT
3.12 Voltage Reflection
Coefficient
definition we have
ZL ,
[m0084]
We now consider the scenario shown in Figure 3.18.
Here a wave arriving from the left along a lossless
transmission line having characteristic impedance Z0
arrives at a termination located at z = 0. The
impedance looking into the termination is ZL , which
may be real-, imaginary-, or complex-valued. The
questions are: Under what circumstances is a
reflection – i.e., a leftward traveling wave – expected,
and what precisely is that wave?
The potential and current of the incident wave are
related by the constant value of Z0 . Similarly, the
potential and current of the reflected wave are related
by Z0 . Therefore, it suffices to consider either
potential or current. Choosing potential, we may
express the incident wave as
Ve + (z) = V0+ e−jβz
(3.75)
where V0+ is determined by the source of the wave,
and so is effectively a “given.” Any reflected wave
must have the form
e−
V
(z) = V0− e+jβz
V
V
Z0
+
VL
(3.77)
(3.78)
(3.79)
where Ie+ (z) and Ie− (z) are the currents associated
with Ve + (z) and Ve − (z), respectively. Since the
voltage and current are related by Z0 , Equation 3.79
may be rewritten as follows:
Ve + (0) Ve − (0)
−
= IeL
Z0
Z0
(3.80)
Evaluating the left sides of Equations 3.78 and 3.80 at
z = 0, we find:
V0+ + V0− = VeL
V0+
−
V0−
Z0
= IeL
(3.81)
(3.82)
Substituting these expressions into Equation 3.77 we
obtain:
V + + V0−
ZL = + 0
(3.83)
V0 /Z0 − V0− /Z0
Solving for V0− we obtain
V0− =
IL
Ve + (0) + Ve − (0) = VeL
Ie+ (0) + Ie− (0) = IeL
(3.76)
Considering the situation at z = 0, note that by
VeL
IeL
where VeL and IeL are the potential across and current
through the termination, respectively. Also, the
potential and current on either side of the z = 0
interface must be equal. Thus,
Z0
Therefore, the problem is solved by determining the
value of V0− given V0+ , Z0 , and ZL .
+
47
ZL − Z0 +
V
ZL + Z0 0
(3.84)
Thus, the answer to the question posed earlier is that
ZL
z
z =0
Figure 3.18: A wave arriving from the left incident on
a termination located at z = 0.
V0− = ΓV0+ , where
Γ,
ZL − Z0
ZL + Z0
(3.85)
(3.86)
The quantity Γ is known as the voltage reflection
coefficient. Note that when ZL = Z0 , Γ = 0 and
therefore V0− = 0. In other words,
CHAPTER 3. TRANSMISSION LINES
48
If the terminating impedance is equal to the characteristic impedance of the transmission line,
then there is no reflection.
If, on the other hand, ZL 6= Z0 , then |Γ| > 0,
V0− = ΓV0+ , and a leftward-traveling reflected wave
exists.
Since ZL may be real-, imaginary-, or
complex-valued, Γ too may be real-, imaginary-, or
complex-valued. Therefore, V0− may be different
from V0+ in magnitude, sign, or phase.
Note also that Γ is not the ratio of I0− to I0+ . The ratio
of the current coefficients is actually −Γ. It is quite
simple to show this with a simple modification to the
above procedure and is left as an exercise for the
student.
Summarizing:
The voltage reflection coefficient Γ, given by
Equation 3.86, determines the magnitude and
phase of the reflected wave given the incident
wave, the characteristic impedance of the transmission line, and the terminating impedance.
[m0085]
We now consider values of Γ that arise for
commonly-encountered terminations.
Matched Load (ZL = Z0 ). In this case, the
termination may be a device with impedance Z0 , or
the termination may be another transmission line
having the same characteristic impedance. When
ZL = Z0 , Γ = 0 and there is no reflection.
Open Circuit. An “open circuit” is the absence of a
termination. This condition implies ZL → ∞, and
subsequently Γ → +1. Since the current reflection
coefficient is −Γ, the reflected current wave is 180◦
out of phase with the incident current wave, making
the total current at the open circuit equal to zero, as
expected.
Short Circuit. “Short circuit” means ZL = 0, and
subsequently Γ = −1. In this case, the phase of Γ is
180◦ , and therefore, the potential of the reflected
wave cancels the potential of the incident wave at the
open circuit, making the total potential equal to zero,
as it must be. Since the current reflection coefficient
is −Γ = +1 in this case, the reflected current wave is
in phase with the incident current wave, and the
magnitude of the total current at the short circuit
non-zero as expected.
Purely Reactive Load. A purely reactive load,
including that presented by a capacitor or inductor,
has ZL = jX where X is reactance. In particular, an
inductor is represented by X > 0 and a capacitor is
represented by X < 0. We find
Γ=
−Z0 + jX
+Z0 + jX
(3.87)
The numerator and denominator have the same
magnitude, so |Γ| = 1. Let φ be the phase of the
denominator (+Z0 + jX). Then, the phase of the
numerator is π − φ. Subsequently, the phase of Γ is
(π − φ) − φ = π − 2φ. Thus, we see that the phase of
Γ is no longer limited to be 0◦ or 180◦ , but can be any
value in between. The phase of reflected wave is
subsequently shifted by this amount.
Other Terminations. Any other termination,
including series and parallel combinations of any
number of devices, can be expressed as a value of ZL
which is, in general, complex-valued. The associated
3.13. STANDING WAVES
49
value of |Γ| is limited to the range 0 to 1. To see this,
note:
ZL − Z0
ZL /Z0 − 1
Γ=
=
(3.88)
ZL + Z0
ZL /Z0 + 1
Note that the smallest possible value of |Γ| occurs
when the numerator is zero; i.e., when ZL = Z0 .
Therefore, the smallest value of |Γ| is zero. The
largest possible value of |Γ| occurs when
ZL /Z0 → ∞ (i.e., an open circuit) or when
ZL /Z0 = 0 (a short circuit); the result in either case
is |Γ| = 1. Thus,
0 ≤ |Γ| ≤ 1
(3.89)
3.13 Standing Waves
[m0086]
A standing wave consists of waves moving in opposite directions. These waves add to make a
distinct magnitude variation as a function of distance that does not vary in time.
To see how this can happen, first consider that an
incident wave V0+ e−jβz , which is traveling in the +z
axis along a lossless transmission line. Associated
with this wave is a reflected wave
V0− e+jβz = ΓV0+ e+jβz , where Γ is the voltage
reflection coefficient. These waves add to make the
total potential
Ve (z) = V0+ e−jβz + ΓV0+ e+jβz
= V0+ e−jβz + Γe+jβz
(3.90)
The magnitude of Ve (z) is most easily found by first
finding |Ve (z)|2 , which is:
Ve (z)Ve ∗ (z)
=|V0+ |2 e−jβz + Γe+jβz
e−jβz + Γe+jβz
∗
=|V0+ |2 e−jβz + Γe+jβz e+jβz + Γ∗ e−jβz
2
=|V0+ |2 1 + |Γ| + Γe+j2βz + Γ∗ e−j2βz (3.91)
Let φ be the phase of Γ; i.e.,
Γ = |Γ| ejφ
(3.92)
Then, continuing from the previous expression:
2
|V0+ |2 1 + |Γ| + |Γ| e+j(2βz+φ) + |Γ| e−j(2βz+φ)
i
h
2
=|V0+ |2 1 + |Γ| + |Γ| e+j(2βz+φ) + e−j(2βz+φ)
(3.93)
The quantity in square brackets can be reduced to a
cosine function using the identity
cos θ =
yielding:
1 jθ
e + e−jθ
2
h
i
2
|V0+ |2 1 + |Γ| + 2 |Γ| cos (2βz + φ)
(3.94)
Recall that this is |Ve (z)|2 . |Ve (z)| is therefore the
square root of the above expression:
q
2
Ve (z) = |V0+ | 1 + |Γ| + 2 |Γ| cos (2βz + φ)
(3.95)
Thus, we have found that the magnitude of the
resulting total potential varies sinusoidally along the
line. This is referred to as a standing wave because
the variation of the magnitude of the phasor resulting
from the interference between the incident and
reflected waves does not vary with time.
We may perform a similar analysis of the current,
leading to:
+ q
|V |
e
= 0
I(z)
Z0
2
1 + |Γ| − 2 |Γ| cos (2βz + φ)
(3.96)
Again we find the result is a standing wave.
Now let us consider the outcome for a few special
cases.
Matched load. When the impedance of the
termination of the transmission line, ZL , is equal to
the characteristic impedance of the transmission line,
Z0 , Γ = 0 and there is no reflection. In this case, the
above expressions reduce to |Ve (z)| = |V0+ | and
e
|I(z)|
= |V0+ |/Z0 , as expected.
Open or Short-Circuit. In this case, Γ = ±1 and
we find:
p
Ve (z) = |V0+ | 2 + 2 cos (2βz + φ)
(3.97)
CHAPTER 3. TRANSMISSION LINES
50
/
V(z)
2.00
Γ=1
+
01234
Γ*,-.
Γ56789
1.75
2 V0
Γ
|V| / V0
1.50
+
V0
1.25
:;
1.00
Γ
0.75
0.50
0.25
z
0.00
z=0
0
'2
#
(a) Potential.
!"
3(
2$
&z 2 [rad]
)
5
2
by Inductiveload (modified)
I(z)
+
2I0
Figure 3.20: Standing waves associated with loads
exhibiting various reflection coefficients. In this figure
the incident wave arrives from the right.
+
I0
2π (β/π) z, so the frequency of variation is β/π and
the period of the variation is π/β. Since β = 2π/λ,
we see that the period of the variation is λ/2.
Furthermore, this is true regardless of the value of Γ.
z
z=0
(b) Current.
Figure 3.19: Standing wave associated with an opencircuit termination at z = 0 (incident wave arrives
from left).
+ p
|V |
e
I(z)
= 0
Z0
2 − 2 cos (2βz + φ)
%
3
(3.98)
where φ = 0 for an open circuit and φ = π for a short
circuit. The result for an open circuit termination is
shown in Figure 3.19(a) (potential) and 3.19(b)
(current). The result for a short circuit termination is
identical except the roles of potential and current are
reversed. In either case, note that voltage maxima
correspond to current minima, and vice versa.
Also note:
The period of the standing wave is λ/2; i.e., onehalf of a wavelength.
This can be confirmed as follows. First, note that the
frequency argument of the cosine function of the
standing wave is 2βz. This can be rewritten as
Mismatched loads. A common situation is that the
termination is neither perfectly-matched (Γ = 0) nor
an open/short circuit (|Γ| = 1). Examples of the
resulting standing waves are shown in Figure 3.20.
Additional Reading:
• “Standing Wave” on Wikipedia.
3.14. STANDING WAVE RATIO
51
3.14 Standing Wave Ratio
10
8
[m0081]
Precise matching of transmission lines to terminations
is often not practical or possible. Whenever a
significant mismatch exists, a standing wave
(Section 3.13) is apparent. The quality of the match is
commonly expressed in terms of the standing wave
ratio (SWR) of this standing wave.
Standing wave ratio (SWR) is defined as the ratio
of the maximum magnitude of the standing wave
to minimum magnitude of the standing wave.
6
<?@
4
2
0
0
0.2
0.4
0.6
0.8
1
Γ
Figure 3.21: Relationship between SWR and |Γ|.
In terms of the potential:
maximum |Ve |
SWR ,
minimum |Ve |
(3.99)
SWR can be calculated using a simple expression,
which we shall now derive. In Section 3.13, we found
that:
q
2
Ve (z) = |V0+ | 1 + |Γ| + 2 |Γ| cos (2βz + φ)
(3.100)
The maximum value occurs when the cosine factor is
equal to +1, yielding:
q
2
+
e
(3.101)
max V = |V0 | 1 + |Γ| + 2 |Γ|
Note that the argument of the square root operator is
2
equal to (1 + |Γ|) ; therefore:
max Ve = |V0+ | (1 + |Γ|)
(3.102)
Similarly, the minimum value is achieved when the
cosine factor is equal to −1, yielding:
q
2
(3.103)
min Ve = |V0+ | 1 + |Γ| − 2 |Γ|
So:
min Ve = |V0+ | (1 − |Γ|)
(3.104)
SWR =
(3.105)
This relationship is shown graphically in Figure 3.21.
Note that SWR ranges from 1 for perfectly-matched
terminations (Γ = 0) to infinity for open- and
short-circuit terminations (|Γ| = 1).
It is sometimes of interest to find the magnitude of the
reflection coefficient given SWR. Solving
Equation 3.105 for |Γ| we find:
|Γ| =
SWR − 1
SWR + 1
(3.106)
SWR is often referred to as the voltage standing wave
ratio (VSWR), although repeating the analysis above
for the current reveals that the current SWR is equal
to potential SWR, so the term “SWR” suffices.
SWR < 2 or so is usually considered a “good match,”
although some applications require SWR < 1.1 or
better, and other applications are tolerant to SWR of 3
or greater.
Example 3.3. Reflection coefficient for various
values of SWR.
What is the reflection coefficient for the
above-cited values of SWR? Using
Equation 3.106, we find:
Therefore:
1 + |Γ|
1 − |Γ|
• SWR = 1.1 corresponds to |Γ| = 0.0476.
CHAPTER 3. TRANSMISSION LINES
52
• SWR = 2.0 corresponds to |Γ| = 1/3.
• SWR = 3.0 corresponds to |Γ| = 1/2.
3.15 Input Impedance of a
Terminated Lossless
Transmission Line
[m0087]
Consider Figure 3.22, which shows a lossless
transmission line being driven from the left and which
is terminated by an impedance ZL on the right. If ZL
is equal to the characteristic impedance Z0 of the
transmission line, then the input impedance Zin will
be equal to ZL . Otherwise Zin depends on both ZL
and the characteristics of the transmission line. In this
section, we determine a general expression for Zin in
terms of ZL , Z0 , the phase propagation constant β,
and the length l of the line.
Using the coordinate system indicated in Figure 3.22,
the interface between source and transmission line is
located at z = −l. Impedance is defined at the ratio of
potential to current, so:
Zin (l) ,
Ve (z = −l)
e = −l)
I(z
(3.107)
e from
Now employing expressions for Ve (z) and I(z)
Section 3.13 with z = −l, we find:
V0+ e+jβl + Γe−jβl
Zin (l) = + +jβl
− Γe−jβl ) /Z0
V0 (e
e+jβl + Γe−jβl
= Z0 +jβl
e
− Γe−jβl
(3.108)
Multiplying both numerator and denominator by
l
Z0
ZL
Zin
z
z =Cl
z =0
Figure 3.22: A transmission line driven by a source on
the left and terminated by an impedance ZL at z = 0
on the right.
3.15. INPUT IMPEDANCE OF A TERMINATED LOSSLESS TRANSMISSION LINE
e−jβl :
Zin (l) = Z0
1 + Γe−j2βl
1 − Γe−j2βl
(3.109)
Recall that Γ in the above expression is:
Γ=
ZL − Z0
ZL + Z0
(3.110)
Summarizing:
Equation 3.109 is the input impedance of a
lossless transmission line having characteristic
impedance Z0 and which is terminated into a load
ZL . The result also depends on the length and
phase propagation constant of the line.
Note that Zin (l) is periodic in l. Since the argument
of the complex exponential factors is 2βl, the
frequency at which Zin (l) varies is β/π; and since
β = 2π/λ, the associated period is λ/2. This is very
useful to keep in mind because it means that all
possible values of Zin (l) are achieved by varying l
over λ/2. In other words, changing l by more than
λ/2 results in an impedance which could have been
obtained by a smaller change in l. Summarizing to
underscore this important idea:
The input impedance of a terminated lossless
transmission line is periodic in the length of the
transmission line, with period λ/2.
Not surprisingly, λ/2 is also the period of the
standing wave (Section 3.13). This is because – once
again – the variation with length is due to the
interference of incident and reflected waves.
Also worth noting is that Equation 3.109 can be
written entirely in terms of ZL and Z0 , since Γ
depends only on these two parameters. Here’s that
version of the expression:
ZL + jZ0 tan βl
(3.111)
Zin (l) = Z0
Z0 + jZL tan βl
This expression can be derived by substituting
Equation 3.110 into Equation 3.109 and is left as an
exercise for the student.
53
Finally, note that the argument βl appearing
Equations 3.109 and 3.111 has units of radians and is
referred to as electrical length. Electrical length can
be interpreted as physical length expressed with
respect to wavelength and has the advantage that
analysis can be made independent of frequency.
CHAPTER 3. TRANSMISSION LINES
[m0088]
Let us now consider the input impedance of a
transmission line that is terminated in an open- or
short-circuit. Such a transmission line is sometimes
referred to as a stub. First, why consider such a
thing? From a “lumped element” circuit theory
perspective, this would not seem to have any
particular application. However, the fact that this
structure exhibits an input impedance that depends on
length (Section 3.15) enables some very useful
applications.
First, let us consider the question at hand: What is the
input impedance when the transmission line is openor short-circuited? For a short circuit, ZL = 0,
Γ = −1, so we find
1 + Γe−j2βl
1 − Γe−j2βl
1 − e−j2βl
= Z0
1 + e−j2βl
0
0
0.25
0.5
length [wavelengths]
0.75
1
(a) Short-circuit termination (ZL = 0).
imaginary part of input impedance
3.16 Input Impedance for Openand Short-Circuit
Terminations
imaginary part of input impedance
54
0
Zin (l) = Z0
(3.112)
Multiplying numerator and denominator by e+jβl we
obtain
e+jβl − e−jβl
Zin (l) = Z0 +jβl
e
+ e−jβl
(3.113)
0
0.25
0.5
length [wavelengths]
0.75
1
(b) Open-circuit termination (ZL → ∞).
Figure 3.23: Input reactance (Im{Zin }) of a stub.
Re{Zin } is always zero.
Now we invoke the following trigonometric identities:
1 +jθ
e
+ e−jθ
2
1 +jθ
sin θ =
e
− e−jθ
j2
cos θ =
(3.114)
(3.115)
Employing these identities, we obtain:
Zin (l) = Z0
j2 (sin βl)
2 (cos βl)
(3.116)
circuit with no transmission line. Also, Zin varies
periodically with increasing length, with period λ/2.
This is precisely as expected from standing wave
theory (Section 3.13). What is of particular interest
now is that as l → λ/4, we see Zin → ∞.
Remarkably, the transmission line has essentially
transformed the short circuit termination into an open
circuit!
For an open circuit termination, ZL → ∞, Γ = +1,
and we find
and finally:
Zin (l) = +jZ0 tan βl
(3.117)
Figure 3.23(a) shows what’s going on. As expected,
Zin = 0 when l = 0, since this amounts to a short
1 + Γe−j2βl
1 − Γe−j2βl
1 + e−j2βl
= Z0
1 − e−j2βl
Zin (l) = Z0
(3.118)
3.17. APPLICATIONS OF OPEN- AND SHORT-CIRCUITED TRANSMISSION LINE STUBS
55
3.17 Applications of Open- and
Short-Circuited
(3.119)
Transmission Line Stubs
Following the same procedure detailed above for the
short-circuit case, we find
Zin (l) = −jZ0 cot βl
Figure 3.23(b) shows the result for open-circuit
termination. As expected, Zin → ∞ for l = 0, and
the same λ/2 periodicity is observed. What is of
particular interest now is that at l = λ/4 we see
Zin = 0. In this case, the transmission line has
transformed the open circuit termination into a short
circuit.
Now taking stock of what we have determined:
The input impedance of a short- or opencircuited lossless transmission line is completely imaginary-valued and is given by Equations 3.117 and 3.119, respectively.
The input impedance of a short- or open-circuited
lossless transmission line alternates between
open- (Zin → ∞) and short-circuit (Zin = 0)
conditions with each λ/4-increase in length.
Additional Reading:
• “Stub (electronics)” on Wikipedia.
[m0145]
The theory of open- and short-circuited transmission
lines – often referred to as stubs – was addressed in
Section 3.16. These structures have important and
wide-ranging applications.
In particular, these structures can be used to replace
discrete inductors and capacitors in certain
applications. To see this, consider the short-circuited
line (Figure 3.23(b) of Section 3.16). Note that each
value of l that is less than λ/4 corresponds to a
particular positive reactance; i.e., the transmission
line “looks” like an inductor. Also note that lengths
between λ/4 and λ/2 result in reactances that are
negative; i.e., the transmission line “looks” like a
capacitor. Thus, it is possible to replace an inductor or
capacitor with a short-circuited transmission line of
the appropriate length. The input impedance of such a
transmission line is identical to that of the inductor or
capacitor at the design frequency. The variation of
reactance with respect to frequency will not be
identical, which may or may not be a concern
depending on the bandwidth and frequency response
requirements of the application. Open-circuited lines
may be used in a similar way.
This property of open- and short-circuited
transmission lines makes it possible to implement
impedance matching circuits (see Section 3.23),
filters, and other devices entirely from transmission
lines, with fewer or no discrete inductors or capacitors
required. Transmission lines do not suffer the
performance limitations of discrete devices at high
frequencies and are less expensive. A drawback of
transmission line stubs in this application is that the
lines are typically much larger than the discrete
devices they are intended to replace.
Example 3.4. Emitter Induction Using
Short-Circuited Line.
In the design of low-noise amplifiers using
CHAPTER 3. TRANSMISSION LINES
56
bipolar transistors in common-emitter
configuration, it is often useful to introduce a
little inductance between the emitter and ground.
This is known as “inductive degeneration,”
“emitter induction,” or sometimes by other
names. It can be difficult to find suitable
inductors, especially for operation in the UHF
band and higher. However, a microstrip line can
be used to achieve the desired inductive
impedance. Determine the length of a stub that
implements a 2.2 nH inductance at 6 GHz using
microstrip line with characteristic impedance
50 Ω and phase velocity 0.6c.
Solution. At the design frequency, the
impedance looking into this section of line from
the emitter should be equal to that of a 2.2 nH
inductor, which is
+jωL = +j2πf L = +j82.9 Ω. The input
impedance of a short-circuited stub of length l
which is grounded (thus, short-circuited) at the
opposite end is +jZ0 tan βl (Section 3.16).
Setting this equal to +j82.9 Ω and noting that
Z0 = 50 Ω, we find that βl ∼
= 1.028 rad. The
phase propagation constant is (Section 3.8):
2πf ∼
ω
=
β=
= 209.4 rad/m
vp
0.6c
(3.120)
Therefore, the length of the microstrip line is
l = (βl) /β ∼
= 4.9 mm.
Additional Reading:
• “Stub (electronics)” on Wikipedia.
3.18 Measurement of
Transmission Line
Characteristics
[m0089]
This section presents a simple technique for
measuring the characteristic impedance Z0 , electrical
length βl, and phase velocity vp of a lossless
transmission line. This technique requires two
measurements: the input impedance Zin when the
transmission line is short-circuited and Zin when the
transmission line is open-circuited.
In Section 3.16, it is shown that the input impedance
Zin of a short-circuited transmission line is
(SC)
Zin
= +jZ0 tan βl
and when a transmission line is terminated in an open
circuit, the input impedance is
(OC)
Zin
= −jZ0 cot βl
Observe what happens when we multiply these results
together:
(SC)
(OC)
Zin · Zin = Z02
(OC)
that is, the product of the measurements Zin and
(SC)
Zin is simply the square of the characteristic
impedance. Therefore
q
(SC)
(OC)
(3.121)
Z0 = Zin · Zin
If we instead divide these measurements, we find
(SC)
Zin
(OC)
Zin
= − tan2 βl
Therefore:
"
tan βl = −
(SC)
Zin
(OC)
Zin
#1/2
(3.122)
If l is known in advance to be less than λ/2, then the
electrical length βl can be determined by taking the
inverse tangent. If l is of unknown length and longer
than λ/2, one must take care to account for the
periodicity of tangent function; in this case, it may not
3.19. QUARTER-WAVELENGTH TRANSMISSION LINE
be possible to unambiguously determine βl. Although
we shall not present the method here, it is possible to
resolve this ambiguity by making multiple
measurements over a range of frequencies.
Once βl is determined, it is simple to determine l
given β, β given l, and then vp . For example, the
phase velocity may be determined by first finding βl
for a known length using the above procedure,
calculating β = (βl) /l, and then vp = ω/β.
57
3.19 Quarter-Wavelength
Transmission Line
[m0091]
Quarter-wavelength sections of transmission line play
an important role in many systems at radio and optical
frequencies. The remarkable properties of open- and
short-circuited quarter-wave line are presented in
Section 3.16 and should be reviewed before reading
further. In this section, we perform a more general
analysis, considering not just open- and short-circuit
terminations but any terminating impedance, and then
we address some applications.
The general expression for the input impedance of a
lossless transmission line is (Section 3.15):
Zin (l) = Z0
1 + Γe−j2βl
1 − Γe−j2βl
(3.123)
Note that when l = λ/4:
2βl = 2 ·
2π λ
· =π
λ 4
Subsequently:
1 + Γe−jπ
1 − Γe−jπ
1−Γ
= Z0
1+Γ
Zin (λ/4) = Z0
(3.124)
Recall that (Section 3.15):
Γ=
ZL − Z0
ZL + Z0
(3.125)
Substituting this expression and then multiplying
numerator and denominator by ZL + Z0 , one obtains
(ZL + Z0 ) − (ZL − Z0 )
(ZL + Z0 ) + (ZL − Z0 )
2Z0
= Z0
2ZL
Zin (λ/4) = Z0
(3.126)
Thus,
Zin (λ/4) =
Z02
ZL
(3.127)
Note that the input impedance is inversely
proportional to the load impedance. For this reason, a
CHAPTER 3. TRANSMISSION LINES
58
l= /4
in the line is λ = 0.6λ0 ∼
= 1.8 cm, and the length
of the line should be l = λ/4 ∼
= 4.5 mm.
The characteristic impedance is given by
Equation 3.128:
√
Z0 = 300 Ω · 50 Ω ∼
(3.129)
= 122.5 Ω
Figure 3.24: Impedance-matching using a quarterwavelength transmission line.
transmission line of length λ/4 is sometimes referred
to as a quarter-wave inverter or simply as a
impedance inverter.
This value would be used to determine the width
of the microstrip line, as discussed in
Section 3.11.
It should be noted that for this scheme to yield a
real-valued characteristic impedance, the product of
the source and load impedances must be a real-valued
number. In particular, this method is not suitable if
Quarter-wave lines play a very important role in RF
ZL has a significant imaginary-valued component and
engineering. As impedance inverters, they have the
matching to a real-valued source impedance is
useful attribute of transforming small impedances into desired. One possible workaround in this case is the
large impedances, and vice-versa – we’ll come back
two-stage strategy shown in Figure 3.25. In this
to this idea later in this section. First, let’s consider
scheme, the load impedance is first transformed to a
how quarter-wave lines are used for impedance
real-valued impedance using a length l1 of
matching. Look what happens when we solve
transmission line. This is accomplished using
Equation 3.127 for Z0 :
Equation 3.123 (quite simple using a numerical
p
search) or using the Smith chart (see “Additional
Z0 = Zin (λ/4) · ZL
(3.128) Reading” at the end of this section). The characteristic
impedance Z01 of this transmission line is not critical
This equation indicates that we may match the load
ZL to a source impedance (represented by Zin (λ/4)) and can be selected for convenience. Normally, the
simply by making the characteristic impedance equal smallest value of l1 is desired. This value will always
to the value given by the above expression and setting be less than λ/4 since Zin (l1 ) is periodic in l1 with
period λ/2; i.e., there are two changes in the sign of
the length to λ/4. The scheme is shown in
the
imaginary component of Zin (l1 ) as l1 is increased
Figure 3.24.
from zero to λ/2. After eliminating the imaginary
component of ZL in this manner, the real component
Example 3.5. 300-to-50 Ω match using an
of the resulting impedance may then be transformed
quarter-wave section of line.
using the quarter-wave matching technique described
earlier in this section.
Design a transmission line segment that matches
300 Ω to 50 Ω at 10 GHz using a quarter-wave
Example 3.6. Matching a patch antenna to
match. Assume microstrip line for which
50 Ω.
propagation occurs with wavelength 60% that of
free space.
A particular patch antenna exhibits a source
impedance of ZA = 35 + j35 Ω. (See
Solution. The line is completely specified given
“Microstrip
antenna” in “Additional Reading” at
its characteristic impedance Z0 and length l. The
the
end
of
this
section for some optional reading
length should be one-quarter wavelength with
on
patch
antennas.)
Interface this antenna to
respect to the signal propagating in the line. The
50
Ω
using
the
technique
described above. For
free-space wavelength λ0 = c/f at 10 GHz is
∼
the
section
of
transmission
line adjacent to the
= 3 cm. Therefore, the wavelength of the signal
3.19. QUARTER-WAVELENGTH TRANSMISSION LINE
D/4
L
(completely
real-valued)
Figure 3.25: Impedance-matching a complex-valued
load impedance using quarter-wavelength transmission line.
patch antenna, use characteristic impedance
Z01 = 50 Ω. Determine the lengths l1 and l2 of
the two segments of transmission line, and the
characteristic impedance Z02 of the second
(quarter-wave) segment.
Solution. The length of the first section of the
transmission line (adjacent to the antenna) is
determined using Equation 3.123:
1 + Γe−j2β1 l1
Z1 (l1 ) = Z01
1 − Γe−j2β1 l1
(3.130)
where β1 is the phase propagation constant for
this section of transmission line and
Γ,
ZA − Z01 ∼
= −0.0059 + j0.4142 (3.131)
ZA + Z01
We seek the value of smallest positive value of
β1 l1 for which the imaginary part of Z1 (l1 ) is
zero. This can determined using a Smith chart
(see “Additional Reading” at the end of this
section) or simply by a few iterations of
trial-and-error. Either way we find
Z1 (β1 l1 = 0.793 rad) ∼
= 120.719 − j0.111 Ω,
which we deem to be close enough to be
acceptable. Note that β1 = 2π/λ, where λ is the
wavelength of the signal in the transmission line.
Therefore
β 1 l1
β 1 l1 ∼
l1 =
=
λ = 0.126λ
β1
2π
(3.132)
59
The length of the second section of the
transmission line, being a quarter-wavelength
transformer, should be l2 = 0.25λ. Using
Equation 3.128, the characteristic impedance
Z02 of this section of line should be
p
Z02 ∼
= (120.719 Ω) (50 Ω) ∼
= 77.7 Ω
(3.133)
Discussion. The total length of the matching
structure is l1 + l2 ∼
= 0.376λ. A patch antenna
would typically have sides of length about
λ/2 = 0.5λ, so the matching structure is nearly
as big as the antenna itself. At frequencies where
patch antennas are commonly used, and
especially at frequencies in the UHF
(300–3000 MHz) band, patch antennas are often
comparable to the size of the system, so it is not
attractive to have the matching structure also
require a similar amount of space. Thus, we
would be motivated to find a smaller matching
structure.
Although quarter-wave matching techniques are
generally effective and commonly used, they have one
important contraindication, noted above – They often
result in structures that are large. That is, any
structure which employs a quarter-wave match will be
at least λ/4 long, and λ/4 is typically large compared
to the associated electronics. Other transmission line
matching techniques – and in particular, single stub
matching (Section 3.23) – typically result in
structures which are significantly smaller.
The impedance inversion property of
quarter-wavelength lines has applications beyond
impedance matching. The following example
demonstrates one such application:
Example 3.7. RF/DC decoupling in transistor
amplifiers.
Transistor amplifiers for RF applications often
receive DC current at the same terminal which
delivers the amplified RF signal, as shown in
Figure 3.26. The power supply typically has a
low output impedance. If the power supply is
directly connected to the transistor, then the RF
CHAPTER 3. TRANSMISSION LINES
60
DC power in
(low )
DC power in
(low E)
G
H/4
RF out
(moderate
RF out
(moderate Z)
F)
Figure 3.26: Use of an inductor to decouple the DC Figure 3.27: Decoupling of DC input power and
input power from the RF output signal at the output of RF output signal at the output of a common-emitter
RF amplifier, using a quarter-wavelength transmission
a common-emitter RF amplifier.
line.
will flow predominantly in the direction of the
power supply as opposed to following the
desired path, which exhibits a higher impedance.
This can be addressed using an inductor in series
with the power supply output. This works
because the inductor exhibits low impedance at
DC and high impedance at RF. Unfortunately,
discrete inductors are often not practical at high
RF frequencies. This is because practical
inductors also exhibit parallel capacitance,
which tends to decrease impedance.
A solution is to replace the inductor with a
transmission line having length λ/4 as shown in
Figure 3.27. A wavelength at DC is infinite, so
the transmission line is essentially transparent to
the power supply. At radio frequencies, the line
transforms the low impedance of the power
supply to an impedance that is very large relative
to the impedance of the desired RF path.
Furthermore, transmission lines on printed
circuit boards are much cheaper than discrete
inductors (and are always in stock!).
Additional Reading:
• “Quarter-wavelength impedance transformer” on
Wikipedia.
• “Smith chart” on Wikipedia.
• “Microstrip antenna” on Wikipedia.
3.20 Power Flow on Transmission
Lines
[m0090]
It is often important to know the power associated
with a wave on a transmission line. The power of the
waves incident upon, reflected by, and absorbed by a
load are each of interest. In this section we shall work
out expressions for these powers and consider some
implications in terms of the voltage reflection
coefficient (Γ) and standing wave ratio (SWR).
Let’s begin by considering a lossless transmission line
that is oriented along the z axis. The time-average
power associated with a sinusoidal wave having
potential v(z, t) and current i(z, t) is
Pav (z) ,
1
T
Z t0 +T
v(z, t) i(z, t) dt
(3.134)
t0
where T , 2π/f is one period of the wave and t0 is
the start time for the integration. Since the
time-average power of a sinusoidal signal does not
change with time, t0 may be set equal to zero without
loss of generality.
Let us now calculate the power of a wave incident
from z < 0 on a load impedance ZL at z = 0. We
may express the associated potential and current as
follows:
v + (z, t) = V0+ cos (ωt − βz + φ)
(3.135)
3.20. POWER FLOW ON TRANSMISSION LINES
V0+
cos (ωt − βz + φ)
(3.136)
Z0
And so the associated time-average power is
Z
1 T +
+
v (z, t) i+ (z, t) dt
Pav
(z) =
T 0
Z
2
V0+
1 T
=
·
cos2 (ωt − βz + φ) dt
Z0
T 0
(3.137)
i+ (z, t) =
Employing a well-known trigonometric identity:
cos2 θ =
1 1
+ cos 2θ
2 2
(3.138)
we may rewrite the integrand as follows
1 1
+ cos (2 [ωt − βz + φ])
2 2
(3.139)
Then integrating over both sides of this quantity
Z T
T
cos2 (ωt − βz + φ) dt = + 0
(3.140)
2
0
cos2 (ωt − βz + φ) =
The second term of the integral is zero because it is
the integral of cosine over two complete periods.
Subsequently, we see that the position dependence
(here, the dependence on z) is eliminated. In other
words, the power associated with the incident wave is
the same for all z < 0, as expected. Substituting into
Equation 3.137 we obtain:
+
Pav
=
V0+
2Z0
2
(3.141)
This is the time-average power associated with the
incident wave, measured at any point z < 0 along the
line.
Equation 3.141 gives the time-average power associated with a wave traveling in a single direction along a lossless transmission line.
Using precisely the same procedure, we find that the
power associated with the reflected wave is
−
Pav
=
ΓV0+
2Z0
2
= |Γ|
2
V0+
2Z0
2
(3.142)
or simply
2
−
+
Pav
= |Γ| Pav
(3.143)
61
Equation 3.143 gives the time-average power
associated with the wave reflected from an
impedance mismatch.
Now, what is the power PL delivered to the load
impedance ZL ? The simplest way to calculate this
power is to use the principle of conservation of
power. Applied to the present problem, this principle
asserts that the power incident on the load must equal
the power reflected plus the power absorbed; i.e.,
+
−
Pav
= Pav
+ PL
(3.144)
Applying the previous equations we obtain:
2
+
PL = 1 − |Γ| Pav
(3.145)
Equations 3.145 gives the time-average power
transferred to a load impedance, and is equal to
the difference between the powers of the incident
and reflected waves.
Example 3.8. How important is it to match
50 Ω to 75 Ω?
Two impedances which commonly appear in
radio engineering are 50 Ω and 75 Ω. It is not
uncommon to find that it is necessary to connect
a transmission line having a 50 Ω characteristic
impedance to a device, circuit, or system having
a 75 Ω input impedance, or vice-versa. If no
attempt is made to match these impedances,
what fraction of the power will be delivered to
the termination, and what fraction of power will
be reflected? What is the SWR?
Solution. The voltage reflection coefficient
going from 50 Ω transmission line to a 75 Ω load
is
75 − 50
= +0.2
Γ=
75 + 50
2
The fraction of power reflected is |Γ| = 0.04,
which is 4%. The fraction of power transmitted
2
is 1 − |Γ| , which is 96%. Going from a 50 Ω
transmission line to a 75 Ω termination changes
only the sign of Γ, and therefore, the fractions of
reflected and transmitted power remain 4% and
CHAPTER 3. TRANSMISSION LINES
62
96%, respectively. In either case (from
Section 3.14):
SWR =
1 + |Γ|
= 1.5
1 − |Γ|
This is often acceptable, but may not be good
enough in some particular applications. Suffice
it to say that it is not necessarily required to use
an impedance matching device to connect 50 Ω
to 75 Ω devices.
3.21 Impedance Matching:
General Considerations
[m0092]
“Impedance matching” refers to the problem of
transforming a particular impedance ZL into a
modified impedance Zin . The problem of impedance
matching arises because it is not convenient, practical,
or desirable to have all devices in a system operate at
the same input and output impedances. Here are just a
few of the issues:
• It is not convenient or practical to market coaxial
cables having characteristic impedance equal to
every terminating impedance that might be
encountered.
• Different types of antennas operate at different
impedances, and the impedance of most
antennas vary significantly with frequency.
• Different types of amplifiers operate most
effectively at different output impedances. For
example, amplifiers operating as current sources
operate most effectively with low output
impedance, whereas amplifiers operating as
voltage sources operate most effectively with
high output impedances.
• Independently of the above issue, techniques for
the design of transistor amplifiers rely on
intentionally mismatching impedances; i.e.,
matching to an impedance different than that
which maximizes power transfer or minimizes
reflection. In other words, various design goals
are met by applying particular impedances to the
input and output ports of the transistor.7
For all of these reasons, electrical engineers
frequently find themselves with the task of
transforming a particular impedance ZL into a
modified impedance Zin .
The reader is probably already familiar with many
approaches to the impedance matching problem that
7 For a concise introduction to this concept, see Chapter 10 of
S.W. Ellingson, Radio Systems Engineering, Cambridge Univ. Press,
2016.
3.22. SINGLE-REACTANCE MATCHING
employ discrete components and do not require
knowledge of electromagnetics.8 To list just a few of
these approaches: transformers, resistive (lossy)
matching, single-reactance matching, and
two-reactance (“L” network) matching. However, all
of these have limitations. Perhaps the most serious
limitations pertain to the performance of discrete
components at high frequencies. Here are just a few
of the most common problems:
• Practical resistors actually behave as ideal
resistors in series with ideal inductors
• Practical capacitors actually behave as ideal
capacitors in series with ideal resistors
• Practical inductors behave as ideal inductors in
parallel with ideal capacitors, and in series with
ideal resistors.
All of this makes the use of discrete components
increasingly difficult with increasing frequency.
One possible solution to these types of problems is to
more precisely model each component, and then to
account for the non-ideal behavior by incorporating
the appropriate models in the analysis and design
process. Alternatively, one may consider ways to
replace particular troublesome components – or, in
some cases, all discrete components – with
transmission line devices. The latter approach is
particularly convenient in circuits implemented on
printed circuit boards at frequencies in the UHF band
and higher, since the necessary transmission line
structures are easy to implement as microstrip lines
and are relatively compact since the wavelength is
relatively small. However, applications employing
transmission lines as components in impedance
matching devices can be found at lower frequencies
as well.
63
3.22 Single-Reactance Matching
[m0093]
An impedance matching structure can be designed
using a section of transmission line combined with a
discrete reactance, such as a capacitor or an inductor.
In the strategy presented here, the transmission line is
used to transform the real part of the load impedance
or admittance to the desired value, and then the
reactance is used to modify the imaginary part to the
desired value. (Note the difference between this
approach and the quarter-wave technique described in
Section 3.19. In that approach, the first transmission
line is used to zero the imaginary part.) There are two
versions of this strategy, which we will now consider
separately.
The first version is shown in Figure 3.28. The purpose
of the transmission line is to transform the load
impedance ZL into a new impedance Z1 for which
Re{Z1 } = Re{Zin }. This can be done by solving the
equation (from Section 3.15)
1 + Γe−j2βl
Re {Z1 } = Re Z0
(3.146)
1 − Γe−j2βl
for l, using a numerical search, or using the Smith
chart.9 The characteristic impedance Z0 and phase
propagation constant β of the transmission line are
independent variables and can be selected for
convenience. Normally, the smallest value of l that
satisfies Equation 3.146 is desired. This value will be
≤ λ/4 because the real part of Z1 is periodic in l with
period λ/4.
After matching the real component of the impedance
in this manner, the imaginary component of Z1 may
then be transformed to the desired value (Im{Zin })
by attaching a reactance Xs in series with the
transmission line input, yielding Zin = Z1 + jXS .
Therefore, we choose
Xs = Im {Zin − Z1 }
(3.147)
The sign of Xs determines whether this reactance is a
capacitor (Xs < 0) or inductor (Xs > 0), and the
value of this component is determined from Xs and
the design frequency.
8 For an overview, see Chapter 9 of S.W. Ellingson, Radio Systems Engineering, Cambridge Univ. Press, 2016.
9 For more about the Smith chart, see “Additional Reading” at the
end of this section.
CHAPTER 3. TRANSMISSION LINES
64
jXS
Z0
I
Zin
ZL
solved for βl (as a step toward finding l) is:
1 + Γe−j2βl
(3.150)
1 = Re
1 − Γe−j2βl
By trial and error (or using the Smith chart if you
prefer) we find βl ∼
= 0.408 rad for the primary
line, yielding Z1 ∼
= 50.0 + j29.0 Ω for the input
impedance after attaching the primary line.
Z1
We may now solve for l as follows: Since
vp = ω/β (Section 3.8), we find
Figure 3.28: Single-reactance matching with a series
reactance.
Design a match consisting of a transmission line
in series with a single capacitor or inductor that
matches a source impedance of 50Ω to a load
impedance of 33.9 + j17.6 Ω at 1.5 GHz. The
characteristic impedance and phase velocity of
the transmission line are 50Ω and 0.6c
respectively.
Solution. From the problem statement:
Zin , ZS = 50 Ω and ZL = 33.9 + j17.6 Ω are
the source and load impedances respectively at
f = 1.5 GHz. The characteristic impedance and
phase velocity of the transmission line are
Z0 = 50 Ω and vp = 0.6c respectively.
The reflection coefficient Γ (i.e., ZL with respect
to the characteristic impedance of the
transmission line) is
ZL − Z0 ∼
= −0.142 + j0.239
ZL + Z0
2πf ∼
ω
=
= 52.360 rad/m
vp
0.6c
(3.151)
Therefore l = (βl) /β ∼
= 7.8 mm.
Example 3.9. Single reactance in series.
Γ,
β=
The impedance of the series reactance should be
jXs ∼
= −j29.0 Ω to cancel the imaginary part of
Z1 . Since the sign of this impedance is negative,
it must be a capacitor. The reactance of a
capacitor is −1/ωC, so it must be true that
−
where here Re {Z1 } = Re {ZS } = 50 Ω. So a
more-specific form of the equation that can be
(3.152)
Thus, we find the series reactance is a capacitor
of value C ∼
= 3.7 pF.
The second version of the single-reactance strategy is
shown in Figure 3.29. The difference in this scheme
is that the reactance is attached in parallel. In this
case, it is easier to work the problem using admittance
(i.e., reciprocal impedance) as opposed to impedance;
this is because the admittance of parallel reactances is
simply the sum of the associated admittances; i.e.,
Yin = Y1 + jBp
(3.148)
The length l of the primary line (that is, the one
that connects the two ports of the matching
structure) is determined using the equation:
1 + Γe−j2βl
(3.149)
Re {Z1 } = Re Z0
1 − Γe−j2βl
1 ∼
= −29.0 Ω
2πf C
(3.153)
where Yin = 1/Zin , Y1 = 1/Z1 , and Bp is the
discrete parallel susceptance; i.e., the imaginary part
of the discrete parallel admittance.
So, the procedure is as follows. The transmission line
is used to transform YL into a new admittance Y1 for
which Re{Y1 } = Re{Yin }. First, we note that
Y1 ,
1 − Γe−j2βl
1
= Y0
Z1
1 + Γe−j2βl
(3.154)
where Y0 , 1/Z0 is characteristic admittance.
Again, the characteristic impedance Z0 and phase
3.22. SINGLE-REACTANCE MATCHING
65
of the transmission line are 50Ω and 0.6c
respectively.
P in
jO p
K0
M
YL
J1
Figure 3.29: Single-reactance matching with a parallel
reactance.
propagation constant β of the transmission line are
independent variables and can be selected for
convenience. In the present problem, we aim to solve
the equation
1 − Γe−j2βl
Re {Y1 } = Re Y0
(3.155)
1 + Γe−j2βl
for the smallest value of l, using a numerical search or
using the Smith chart. After matching the real
component of the admittances in this manner, the
imaginary component of the resulting admittance may
then be transformed to the desired value by attaching
the susceptance Bp in parallel with the transmission
line input. Since we desire jBp in parallel with Y1 to
be Yin , the desired value is
Bp = Im {Yin − Y1 }
(3.156)
The sign of Bp determines whether this is a capacitor
(Bp > 0) or inductor (Bp < 0), and the value of this
component is determined from Bp and the design
frequency.
In the following example, we address the same
problem raised in Example 3.9, now using the parallel
reactance approach:
Example 3.10. Single reactance in parallel.
Design a match consisting of a transmission line
in parallel with a single capacitor or inductor
that matches a source impedance of 50Ω to a
load impedance of 33.9 + j17.6 Ω at 1.5 GHz.
The characteristic impedance and phase velocity
Solution. From the problem statement:
Zin , ZS = 50 Ω and ZL = 33.9 + j17.6 Ω are
the source and load impedances respectively at
f = 1.5 GHz. The characteristic impedance and
phase velocity of the transmission line are
Z0 = 50 Ω and vp = 0.6c respectively.
The reflection coefficient Γ (i.e., ZL with respect
to the characteristic impedance of the
transmission line) is
Γ,
ZL − Z0 ∼
= −0.142 + j0.239
ZL + Z0
(3.157)
The length l of the primary line (that is, the one
that connects the two ports of the matching
structure) is the solution to:
1 − Γe−j2βl
(3.158)
Re {Y1 } = Re Y0
1 + Γe−j2βl
where here Re {Y1 } = Re {1/ZS } = 0.02 mho
and Y0 = 1/Z0 = 0.02 mho. So the equation to
be solved for βl (as a step toward finding l) is:
1 − Γe−j2βl
(3.159)
1 = Re
1 + Γe−j2βl
By trial and error (or the Smith chart) we find
βl ∼
= 0.126 rad for the primary line, yielding
Y1 ∼
= 0.0200 − j0.0116 mho for the input
admittance after attaching the primary line.
We may now solve for l as follows: Since
vp = ω/β (Section 3.8), we find
β=
2πf ∼
ω
=
= 52.360 rad/m
vp
0.6c
(3.160)
Therefore, l = (βl) /β ∼
= 2.4 mm.
The admittance of the parallel reactance should
be jBp ∼
= +j0.0116 mho to cancel the
imaginary part of Y1 . The associated impedance
is 1/jBp ∼
= −j86.3 Ω. Since the sign of this
impedance is negative, it must be a capacitor.
The reactance of a capacitor is −1/ωC, so it
CHAPTER 3. TRANSMISSION LINES
66
3.23 Single-Stub Matching
must be true that
−
1 ∼
= −86.3 Ω
2πf C
(3.161)
[m0094]
Thus, we find the parallel reactance is a
capacitor of value C ∼
= 1.2 pF.
In Section 3.22, we considered impedance matching
schemes consisting of a transmission line combined
with a reactance which is placed either in series or in
parallel with the transmission line. In many problems,
Comparing this result to the result from the series
the required discrete reactance is not practical
reactance method (Example 3.9), we see that the
necessary length of transmission line is much shorter, because it is not a standard value, or because of
non-ideal behavior at the desired frequency (see
which is normally a compelling advantage. The
Section 3.21 for more about this), or because one
tradeoff is that the parallel capacitance is much
smaller and an accurate value may be more difficult to might simply wish to avoid the cost and logistical
issues associated with an additional component.
achieve.
Whatever the reason, a possible solution is to replace
the discrete reactance with a transmission line “stub”
Additional Reading:
– that is, a transmission line which has been open- or
short-circuited. Section 3.16 explains how a stub can
• “Smith chart” on Wikipedia.
replace a discrete reactance. Figure 3.30 shows a
practical implementation of this idea implemented in
microstrip. This section explains the theory, and we’ll
return to this implementation at the end of the section.
Figure 3.31 shows the scheme. This scheme is usually
implemented using the parallel reactance approach, as
depicted in the figure. Although a series reactance
scheme is also possible in principle, it is usually not
as convenient. This is because most transmission
lines use one of their two conductors as a local datum;
e.g., the ground plane of a printed circuit board for
microstrip line is tied to ground, and the outer
conductor (“shield”) of a coaxial cable is usually tied
to ground. This is contrast to a discrete reactance
X[\] ^_`a
bcde
QRTUVWtor
Spinningspark CC BY SA 3.0
Figure 3.30: A practical implementation of a singlestub impedance match using microstrip transmission
line. Here, the stub is open-circuited.
3.23. SINGLE-STUB MATCHING
67
lm n
without some kind of remediation to prevent a short
circuit for DC would certainly be a bad idea.
i 01
k1
o in
fL
In the following example we address the same
problem raised in Section 3.22 (Examples 3.9 and
3.10), now using the single-stub approach:
stu
b
Example 3.11. Single stub matching.
h1
stub characteristics: q 02
r2 2
stub may be open- or short-circuited
Figure 3.31: Single-stub matching.
(such as a capacitor or inductor), which does not
require that either of its terminals be tied to ground.
This issue is avoided in the parallel-attached stub
because the parallel-attached stub and the
transmission line to which it is attached both have one
terminal at ground.
The single-stub matching procedure is essentially the
same as the single parallel reactance method, except
the parallel reactance is implemented using a short- or
open-circuited stub as opposed a discrete inductor or
capacitor. Since parallel reactance matching is most
easily done using admittances, it is useful to express
Equations 3.117 and 3.119 (input impedance of an
open- and short-circuited stub, respectively, from
Section 3.16) in terms of susceptance:
Bp = −Y02 cot (β2 l2 ) short-circuited stub (3.162)
Bp = +Y02 tan (β2 l2 ) open-circuited stub (3.163)
As in the main line, the characteristic impedance
Z02 = 1/Y02 is an independent variable and is chosen
for convenience.
A final question is when should you use a
short-circuited stub, and when should you use an
open-circuited stub? Given no other basis for
selection, the termination that yields the shortest stub
is chosen. An example of an “other basis for
selection” that frequently comes up is whether DC
might be present on the line. If DC is present with the
signal of interest, then a short circuit termination
Design a single-stub match that matches a
source impedance of 50Ω to a load impedance of
33.9 + j17.6 Ω. Use transmission lines having
characteristic impedances of 50Ω throughout,
and leave your answer in terms of wavelengths.
Solution. From the problem statement:
Zin , ZS = 50 Ω and ZL = 33.9 + j17.6 Ω are
the source and load impedances respectively.
Z0 = 50 Ω is the characteristic impedance of the
transmission lines to be used. The reflection
coefficient Γ (i.e., ZL with respect to the
characteristic impedance of the transmission
line) is
Γ,
ZL − Z0 ∼
= −0.142 + j0.239
ZL + Z0
(3.164)
The length l1 of the primary line (that is, the one
that connects the two ports of the matching
structure) is the solution to the equation (from
Section 3.22):
1 − Γe−j2β1 l1
(3.165)
Re {Y1 } = Re Y01
1 + Γe−j2β1 l1
where here Re {Y1 } = Re {1/ZS } = 0.02 mho
and Y01 = 1/Z0 = 0.02 mho. Also note
l1
2π
l1 = 4π
2β1 l1 = 2
(3.166)
λ
λ
where λ is the wavelength in the transmission
line. So the equation to be solved for l1 is:
1 − Γe−j4πl1 /λ
(3.167)
1 = Re
1 + Γe−j4πl1 /λ
By trial and error (or using the Smith chart; see
“Additional Reading” at the end of this section)
we find l1 ∼
= 0.020λ for the primary line,
CHAPTER 3. TRANSMISSION LINES
68
yielding Y1 ∼
= 0.0200 − j0.0116 mho for the
input admittance after attaching the primary line.
We now seek the shortest stub having an input
admittance of ∼
= +j0.0116 mho to cancel the
imaginary part of Y1 . For an open-circuited stub,
we need
Bp = +Y0 tan 2πl2 /λ ∼
= +j0.0116 mho
(3.168)
The smallest value of l2 for which this is true is
∼
= 0.084λ. For a short-circuited stub, we need
Bp = −Y0 cot 2πl2 /λ ∼
= +j0.0116 mho
(3.169)
The smallest positive value of l2 for which this is
true is ∼
= 0.334λ; i.e., much longer. Therefore,
we choose the open-circuited stub with
l2 ∼
= 0.084λ. Note the stub is attached in parallel
at the source end of the primary line.
Single-stub matching is a very common method for
impedance matching using microstrip lines at
frequences in the UHF band (300-3000 MHz) and
above. In Figure 3.30, the top (visible) traces
comprise one conductor, whereas the ground plane
(underneath, so not visible) comprises the other
conductor. The end of the stub is not connected to the
ground plane, so the termination is an open circuit. A
short circuit termination is accomplished by
connecting the end of the stub to the ground plane
using a via; that is, a plated-through that electrically
connects the top and bottom layers.
Additional Reading:
• “Stub (electronics)” on Wikipedia.
• “Smith chart” on Wikipedia.
• “Via (electronics)” on Wikipedia.
[m0151]
3.23. SINGLE-STUB MATCHING
Image Credits
Fig. 3.1: Dmitry G, https://en.wikipedia.org/wiki/File:Mastech test leads.JPG,
public domain.
Fig. 3.2: c Tkgd2007, https://commons.wikimedia.org/wiki/File:Coaxial cable cutaway.svg,
CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/).
Minor modifications from the original.
Fig. 3.3: c SpinningSpark, https://en.wikipedia.org/wiki/File:Microstrip structure.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Minor modifications from the original.
Fig. 3.6: c Averse, https://en.wikipedia.org/wiki/File:Diplexer1.jpg,
CC BY SA 2.0 Germany (https://creativecommons.org/licenses/by-sa/2.0/de/deed.en).
Fig. 3.7: c BigRiz, https://en.wikipedia.org/wiki/File:Fibreoptic.jpg,
CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en).
Fig. 3.8: c Omegatron, https://commons.wikimedia.org/wiki/File:Transmission line symbols.svg,
CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en).
Modified.
Fig. 3.10: c Omegatron, https://commons.wikimedia.org/wiki/File:Transmission line element.svg,
CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en).
Modified.
Fig. 3.11: c Omegatron, https://commons.wikimedia.org/wiki/File:Transmission line element.svg,
CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en).
Modified.
Fig. 3.15: c Arj, https://en.wikipedia.org/wiki/File:RG-59.jpg,
CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en).
Fig. 3.16: c 7head7metal7, https://commons.wikimedia.org/wiki/File:Microstrip scheme.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Minor modifications from the original.
Fig. 3.20: by Inductiveload, https://commons.wikimedia.org/wiki/File:Standing Wave Ratio.svg,
public domain.
Modifications from the original.
Fig. 3.30: c Spinningspark, https://commons.wikimedia.org/wiki/File:Stripline stub matching.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Minor modifications from the original: curly braces deleted.
69
Chapter 4
Vector Analysis
4.1 Vector Arithmetic
z
[m0006]
A vector is a mathematical object that has both a
scalar part (i.e., a magnitude and possibly a phase), as
well as a direction. Many physical quantities are best
described as vectors. For example, the rate of
movement through space can be described as speed;
i.e., as a scalar having SI base units of m/s. However,
this quantity is more completely described as velocity;
i.e., as a vector whose scalar part is speed and
direction indicates the direction of movement.
Similarly, force is a vector whose scalar part indicates
magnitude (SI base units of N), and direction
indicates the direction in which the force is applied.
Electric and magnetic fields are also best described as
vectors.
In mathematical notation, a real-valued vector A is
said to have a magnitude A = |A| and direction â
such that
A = Aâ
(4.1)
where â is a unit vector (i.e., a real-valued vector
having magnitude equal to one) having the same
direction as A. If a vector is complex-valued, then A
is similarly complex-valued.
Cartesian Coordinate System. Fundamentals of
vector arithmetic are most easily grasped using the
Cartesian coordinate system. This system is shown in
Figure 4.1. Note carefully the relative orientation of
the x, y, and z axes. This orientation is important. For
example, there are two directions that are
perpendicular to the z = 0 plane (in which the x- and
y-axes lie), but the +z axis is specified to be one of
y
stuwx{
x
c K. Kikkeri CC BY SA 4.0
Figure 4.1: Cartesian coordinate system.
these in particular.
Position-Fixed vs. Position-Free Vectors. It is often
convenient to describe a position in space as a vector
for which the magnitude is the distance from the
origin of the coordinate system and for which the
direction is measured from the origin toward the
position of interest. This is shown in Figure 4.2.
These position vectors are “position-fixed” in the
sense that they are defined with respect to a single
point in space, which in this case is the origin.
Position vectors can also be defined as vectors that are
defined with respect to some other point in space, in
which case they are considered position-fixed to that
position.
Position-free vectors, on the other hand, are not
defined with respect to a particular point in space. An
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
4.1. VECTOR ARITHMETIC
71
z
z
z
y
x
r1
’
|
r}
x
c K. Kikkeri CC BY SA 4.0
x
c K. Kikkeri CC BY SA 4.0
Figure 4.4: Basis vectors in the Cartesian coordinate
system.
Figure 4.2: Position vectors. The vectors r1 and r1
are position-fixed and refer to particular locations.
v‡
~€‚ƒ„ †
v‘
ˆ‰Š‹ŒŽ 
c K. Kikkeri CC BY SA 4.0
Figure 4.3: Two particles exhibiting the same velocity. In this case, the velocity vectors v1 and v2 are
position-free and equal.
example is shown in Figure 4.3. Particles 1 m apart
may both be traveling at 2 m/s in the same direction.
In this case, the velocity of each particle can be
described using the same vector, even though the
particles are located at different points in space.
Position-free vectors are said to be equal if they have
the same magnitudes and directions. Position-fixed
vectors, on the other hand, must also be referenced to
the same position (e.g., the origin) to be considered
equal.
Basis Vectors. Each coordinate system is defined in
terms of three basis vectors which concisely describe
all possible ways to traverse three-dimensional space.
A basis vector is a position-free unit vector that is
perpendicular to all other basis vectors for that
coordinate system. The basis vectors x̂, ŷ, and ẑ of
the Cartesian coordinate system are shown in
Figure 4.4. In this notation, x̂ indicates the direction
in which x increases most rapidly, ŷ indicates the
direction in which y increases most rapidly, and ẑ
indicates the direction in which z increases most
rapidly. Alternatively, you might interpret x̂, ŷ, and ẑ
as unit vectors that are parallel to the x-, y-, and
z-axes and point in the direction in which values
along each axis increase.
Vectors in the Cartesian Coordinate System. In
Cartesian coordinates, we may describe any vector A
as follows:
A = x̂Ax + ŷAy + ẑAz
(4.2)
where Ax , Ay , and Az are scalar quantities describing
the components of A in each of the associated
directions, as shown in Figure 4.5. This description
makes it clear that the magnitude of A is:
|A| =
q
A2x + A2y + A2z
(4.3)
and therefore, we can calculate the associated unit
CHAPTER 4. VECTOR ANALYSIS
72
z
r ¡ £ r¤ r¢
rž
•z
A
rŸ
“
–—
˜™š›œ
c K. Kikkeri CC BY SA 4.0
Figure 4.6: Relative position (distance and direction)
between locations identified by their position vectors.
”x
x
c K. Kikkeri CC BY SA 4.0
Figure 4.5: Components of a vector A in the Cartesian coordinate system.
A
A
=q
|A|
2
A + A2 + A2
x
y
z
−1/2
= x̂Ax A2x + A2y + A2z
−1/2
+ ŷAy A2x + A2y + A2z
−1/2
+ ẑAz A2x + A2y + A2z
(4.4)
Vector Addition and Subtraction. It is common to
add and subtract vectors. For example, vectors
describing two forces A and B applied to the same
point can be described as a single force vector C that
is the sum of A and B; i.e., C = A + B. This
addition is quite simple in the Cartesian coordinate
system:
C=A+B
= (x̂Ax + ŷAy + ẑAz ) + (x̂Bx + ŷBy + ẑBz )
= x̂ (Ax + Bx ) + ŷ (Ay + By ) + ẑ (Az + Bz )
(4.5)
In other words, the x̂ component of C is the sum of
the x̂ components of A and B, and similarly for ŷ
and ẑ. From the above example, it is clear that vector
addition is commutative; i.e.,
A+B=B+A
D=A−B
= (x̂Ax + ŷAy + ẑAz ) − (x̂Bx + ŷBy + ẑBz )
= x̂ (Ax − Bx ) + ŷ (Ay − By ) + ẑ (Az − Bz )
(4.7)
vector as
â =
In other words, vectors may be added in any order.
Vector subtraction is defined similarly:
(4.6)
In other words, the x̂ component of D is the
difference of the x̂ components of A and B, and
similarly for ŷ and ẑ. Like scalar subtraction, vector
subtraction is not commutative.
Relative Positions and Distances. A common task in
vector analysis is to describe the position of one point
in space relative to a different point in space. Let us
identify those two points using the position vectors r1
and r2 , as indicated in Figure 4.6. We may identify a
third vector r12 as the position of r2 relative to r1 :
r12 , r2 − r1
(4.8)
Now |r12 | is the distance between these points, and
r12 / |r12 | is a unit vector indicating the direction to r2
from r1 .
Example 4.1. Direction and distance between
positions.
Consider two positions, identified using the
position vectors r1 = 2x̂ + 3ŷ + 1ẑ and
r2 = 1x̂ − 2ŷ + 3ẑ, both expressed in units of
meters. Find the direction vector that points
4.1. VECTOR ARITHMETIC
73
from r1 to r2 , the distance between these points,
and the associated unit vector.
Solution. The vector that points from r1 to r2 is
R = r2 − r1
= (1 − 2)x̂ + (−2 − 3)ŷ + (3 − 1)ẑ
= −x̂ − 5ŷ + 2ẑ
(4.9)
The distance between r1 and r2 is simply the
magnitude of this vector:
q
2
2
2
|R| = (−1) + (−5) + (2) ∼
= 5.48 m
(4.10)
The unit vector R̂ is simply R normalized to
have unit magnitude:
R̂ = R/ |R|
∼
= (−x̂ − 5ŷ + 2ẑ) /5.48
∼
= −0.182x̂ − 0.913ŷ + 0.365ẑ
c K. Kikkeri CC BY SA 4.0
(4.11)
Multiplication of a Vector by a Scalar. Let’s say a
particular force is specified by a vector F. What is the
new vector if this force is doubled? The answer is
simply 2F – that is, twice the magnitude applied in
the same direction. This is an example of scalar
multiplication of a vector. Generalizing, the product
of the scalar α and the vector A is simply αA.
Scalar (“Dot”) Product of Vectors. Another
common task in vector analysis is to determine the
similarity in the direction in which two vectors point.
In particular, it is useful to have a metric which, when
applied to the vectors A = âA and B = b̂B, has the
following properties (see Figure 4.7):
• If A is perpendicular to B, the result is zero.
• If A and B point in the same direction, the result
is AB.
• If A and B point in opposite directions, the
result is −AB.
• Results intermediate to these conditions depend
on the angle ψ between A and B, measured as if
A and B were arranged “tail-to-tail” as shown in
Figure 4.8.
Figure 4.7: Special cases of the dot product.
In vector analysis, this operator is known as the scalar
product (not to be confused with scalar
multiplication) or the dot product. The dot product is
written A · B and is given in general by the
expression:
A · B = AB cos ψ
(4.12)
Note that this expression yields the special cases
previously identified, which are ψ = π/2, ψ = 0, and
ψ = π, respectively. The dot product is commutative;
i.e.,
A·B=B·A
(4.13)
The dot product is also distributive; i.e.,
A · (B + C) = A · B + A · C
(4.14)
The dot product has some other useful properties. For
example, note:
A · A = (x̂Ax + ŷAy + ẑAz ) · (x̂Ax + ŷAy + ẑAz )
= x̂ · x̂A2x + x̂ · ŷAx Ay + x̂ · ẑAx Az
+ ŷ · x̂Ax Ay + ŷ · ŷA2y + ŷ · ẑAy Az
+ ẑ · x̂Ax Az + ẑ · ŷAy Az + ẑ · ẑA2z
(4.15)
which looks pretty bad until you realize that
x̂ · x̂ = ŷ · ŷ = ẑ · ẑ = 1
(4.16)
CHAPTER 4. VECTOR ANALYSIS
74
A
seek. From Equation 4.19:
C · D = Cx Dx + Cy Dy + Cz Dz
= 2 · 3 + 3 · (−2) + 1 · 2 m
2
B
¥
(4.21)
2
(4.22)
=2m
(4.23)
q
Cx2 + Cy2 + Cz2 ∼
= 3.742 m
(4.24)
Dx2 + Dy2 + Dz2 ∼
= 4.123 m
(4.25)
also
C=
D=
c K. Kikkeri CC BY SA 4.0
so
Figure 4.8: Calculation of the dot product.
and any other dot product of basis vectors is zero.
Thus, the whole mess simplifies to:
A · A = A2x + A2y + A2z
(4.17)
This is the square of the magnitude of A, so we have
discovered that
2
A · A = |A| = A2
(4.18)
q
C·D ∼
(4.26)
= 0.130
CD
Taking the inverse cosine, we find ψ = 82.6◦ .
cos ψ =
Cross Product. The cross product is a form of vector
multiplication that results in a vector that is
perpendicular to both of the operands. The definition
is as follows:
A × B = n̂AB sin ψAB
(4.27)
As shown in Figure 4.9, the unit vector n̂ is
determined by the “right hand rule.” Using your right
hand, curl your fingers to traverse the angle ψAB
A · B = (x̂Ax + ŷAy + ẑAz ) · (x̂Bx + ŷBy + ẑBz ) beginning at A and ending at B, and then n̂ points in
the direction of your fully-extended thumb.
= Ax Bx + Ay By + Az Bz
(4.19)
It should be apparent that the cross product is not
This is a particularly easy way to calculate the dot
commutative but rather is anticommutative; that is,
product, since it eliminates the problem of
A × B = −B × A
(4.28)
determining the angle ψ. In fact, an easy way to
calculate ψ given A and B is to first calculate the dot
You can confirm this for yourself using either
product using Equation 4.19 and then use the result to
Equation 4.27 or by applying the right-hand rule.
solve Equation 4.12 for ψ.
Applying the same principles to the dot product of
potentially different vectors A and B, we find:
The cross product is distributive:
Example 4.2. Angle between two vectors.
Consider the position vectors C = 2x̂ + 3ŷ + 1ẑ
and D = 3x̂ − 2ŷ + 2ẑ, both expressed in units
of meters. Find the angle between these vectors.
Solution. From Equation 4.12
C · D = CD cos ψ
(4.20)
where C = |C|, D = |D|, and ψ is the angle we
A × (B + C) = A × B + A × C
(4.29)
There are two useful special cases of the cross
product that are worth memorizing. The first is the
cross product of a vector with itself, which is zero:
A×A=0
(4.30)
The second is the cross product of vectors that are
perpendicular; i.e., for which ψAB = π/2. In this
4.1. VECTOR ARITHMETIC
75
It is typically awkward to “manually” determine n̂ in
Equation 4.27. However, in Cartesian coordinates the
cross product may be calculated as:
n
A×B=
B
¦
AB
x̂ (Ay Bz − Az By )
+ ŷ (Az Bx − Ax Bz )
+ ẑ (Ax By − Ay Bx )
(4.36)
This may be easier to remember as a matrix
determinant:
x̂
A × B = Ax
Bx
A
ŷ
Ay
By
ẑ
Az
Bz
(4.37)
c K. Kikkeri CC BY SA 4.0
Similar expressions are available for other coordinate
systems.
Figure 4.9: The cross product A × B.
z
Vector analysis routinely requires expressions
involving both dot products and cross products in
different combinations. Often, these expressions may
be simplified, or otherwise made more convenient,
using the vector identities listed in Appendix B.3.
y
x
Additional Reading:
Figure 4.10: Cross products among basis vectors in
the Cartesian system. The cross product of any two
basis vectors is the third basis vector when the order
of operands is counter-clockwise, as shown in the diagram, and is −1 times the third basis vector when the
order of operands is clockwise with respect to the arrangement in the diagram.
case:
A × B = n̂AB
(4.31)
Using these principles, note:
x̂ × x̂ = ŷ × ŷ = ẑ × ẑ = 0
(4.32)
x̂ × ŷ = ẑ
(4.33)
ŷ × ẑ = x̂
(4.34)
ẑ × x̂ = ŷ
(4.35)
whereas
A useful diagram that summarizes these relationships
is shown in Figure 4.10.
• “Vector field” on Wikipedia.
• “Vector algebra” on Wikipedia.
CHAPTER 4. VECTOR ANALYSIS
76
4.2 Cartesian Coordinates
[m0004]
The Cartesian coordinate system is introduced in
Section 4.1. Concepts described in that section – i.e.,
the dot product and cross product – are described in
terms of the Cartesian system. In this section, we
identify some additional features of this system that
are useful in subsequent work and also set the stage
for alternative systems; namely the cylindrical and
spherical coordinate systems.
would have required two basis directions to be
considered – namely, both x and y. In this case,
another system – namely, cylindrical coordinates
(Section 4.3) – minimizes the number of varying
coordinates (to just one, which is φ).
Integration Over Area. Now we ask the question,
what is the integral of some vector field A over some
surface S? The answer is
Z
A · ds
(4.41)
S
Integration Over Length. Consider a vector field
A = x̂A(r), where r is a position vector. What is the
integral of A over some curve C through space? To
answer this question, we first identify a
differential-length segment of the curve. Note that
this segment of the curve can be described as
We refer to ds as the differential surface element,
which is a vector having magnitude equal to the
differential area ds, and is normal (perpendicular) to
each point on the surface. There are actually two such
directions. We’ll return to clear up the ambiguity in a
moment. Now, as an example, if A = ẑ and S is the
surface bounded by x1 ≤ x ≤ x2 , y1 ≤ y ≤ y2 , then
dl = x̂dx + ŷdy + ẑdz
ds = ẑ dx dy
(4.38)
The contribution to the integral for that segment of
the curve is simply A · dl. We integrate to obtain the
result; i.e.,
Z
A · dl
(4.39)
C
For example, if A = x̂A0 (i.e., A(r) is a constant)
and if C is a straight line from x = x1 and x = x2
along some constant y and z, then dl = x̂dx,
A · dl = A0 dx, and subsequently the above integral is
Z x2
A0 dx = A0 (x2 − x1 )
(4.40)
x1
In particular, notice that if A0 = 1, then this integral
gives the length of C. Although the formalism seems
unnecessary in this simple example, it becomes very
useful when integrating over paths that vary in more
than one direction and with more complicated
integrands.
Note that the Cartesian system was an appropriate
choice for preceding example because this allowed
two of the three basis directions (i.e., y and z) to be
immediately eliminated from the calculation. Said
differently, the preceding example is expressed with
the minimum number of varying coordinates in the
Cartesian system. Here’s a counter-example. If C had
been a circle in the z = 0 plane, then the problem
(4.42)
since dxdy is differential surface area in the z = 0
plane and ẑ is normal to the z = 0 plane. So
A · ds = dxdy, and subsequently the integral in
Equation 4.41 becomes
Z x 2 Z y2
dx dy = (x2 − x1 ) (y2 − y1 )
(4.43)
x1
y1
Note that this has turned out to be a calculation of
area.
Once again, we see the Cartesian system was an
appropriate choice for this example because this
choice minimizes the number of varying coordinates;
in the above example, the surface of integration is
described by a constant value of z with variable
values of x and y. If the surface had instead been a
cylinder or a sphere, not only would all three basis
directions be variable, but also the surface normal
would be variable, making the problem dramatically
more complicated.
Now let’s return to the issue of the direction of ds.
We chose +ẑ, but why not choose −ẑ – also a normal
to the surface – as this direction? To answer simply,
the resulting area would be negative. “Negative area”
is the expected (“positive”) area, except with respect
to the opposite-facing normal vector. In the present
problem, the sign of the area is not important, but in
4.3. CYLINDRICAL COORDINATES
77
some problems this sign becomes important. One
example of a class of problems for which the sign of
area is important is when the quantity of interest is a
flux. If A were a flux density, then the integration over
area that we just performed indicates the magnitude
and direction of flux, and so the direction chosen for
ds defines the direction of positive flux. Section 2.4
describes the electric field in terms of a flux (i.e., as
electric flux density D), in which case positive flux
flows away from a positively-charged source.
Integration Over Volume. Another common task in
vector analysis is integration of some quantity over a
volume. Since the procedure is the same for scalar or
vector quantities, we shall consider integration of a
scalar quantity A(r) for simplicity. First, we note that
the contribution from a differential volume element
dv = dx dy dz
(4.44)
is A(r) dv, so the integral over the volume V is
Z
A(r) dv
(4.45)
V
4.3 Cylindrical Coordinates
[m0096]
Cartesian coordinates (Section 4.2) are not convenient
in certain cases. One of these is when the problem has
cylindrical symmetry. For example, in the Cartesian
coordinate system, the cross-section of a cylinder
concentric with the z-axis requires two coordinates to
describe: x and y. However, this cross section can be
described using a single parameter – namely the
radius – which is ρ in the cylindrical coordinate
system. This results in a dramatic simplification of
the mathematics in some applications.
The cylindrical system is defined with respect to the
Cartesian system in Figure 4.11. In lieu of x and y,
the cylindrical system uses ρ, the distance measured
from the closest point on the z axis,1 and φ, the angle
measured in a plane of constant z, beginning at the
+x axis (φ = 0) with φ increasing toward the +y
direction.
The basis vectors in the cylindrical system are ρ̂, φ̂,
and ẑ. As in the Cartesian system, the dot product of
like basis vectors is equal to one, and the dot product
of unlike basis vectors is equal to zero. The cross
For example, if A(r) = 1 and V is a cube bounded by
x1 ≤ x ≤ x2 , y1 ≤ y ≤ y2 , and z1 ≤ z ≤ z2 , then
the above integral becomes
1 Note that some textbooks use “r” in lieu of ρ for this coordinate.
Z x 2 Z y2 Z z 2
dx dy dz = (x2 − x1 ) (y2 − y1 ) (z2 − z1 )
x1
y1
z
z1
(4.46)
i.e., this is a calculation of volume.
z
¨
The Cartesian system was an appropriate choice for
this example because V is a cube, which is easy to
describe in Cartesian coordinates and relatively
difficult to describe in any other coordinate system.
©
z
Additional Reading:
ª
• “Cartesian coordinate system” on Wikipedia.
§
«
x
c K. Kikkeri CC BY SA 4.0
Figure 4.11: Cylindrical coordinate system and associated basis vectors.
CHAPTER 4. VECTOR ANALYSIS
78
z
·
x̂
ŷ
ẑ
­
¬
φ̂
− sin φ
cos φ
0
ẑ
0
0
1
Table 4.1: Dot products between basis vectors in the
cylindrical and Cartesian coordinate systems.
Figure 4.12: Cross products among basis vectors in the
cylindrical system. (See Figure 4.10 for instructions
on the use of this diagram.)
products of basis vectors are as follows:
ρ̂ × φ̂ = ẑ
(4.47)
φ̂ × ẑ = ρ̂
(4.48)
ẑ × ρ̂ = φ̂
(4.49)
A useful diagram that summarizes these relationships
is shown in Figure 4.12.
The cylindrical system is usually less useful than the
Cartesian system for identifying absolute and relative
positions. This is because the basis directions depend
on position. For example, ρ̂ is directed radially
outward from the ẑ axis, so ρ̂ = x̂ for locations along
the x-axis but ρ̂ = ŷ for locations along the y axis.
Similarly, the direction φ̂ varies as a function of
position. To overcome this awkwardness, it is
common to set up a problem in cylindrical
coordinates in order to exploit cylindrical symmetry,
but at some point to convert to Cartesian coordinates.
Here are the conversions:
x = ρ cos φ
(4.50)
y = ρ sin φ
(4.51)
and z is identical in both systems. The conversion
from Cartesian to cylindrical is as follows:
p
(4.52)
ρ = x2 + y 2
φ = arctan (y, x)
ρ̂
cos φ
sin φ
0
(4.53)
where arctan is the four-quadrant inverse tangent
function; i.e., arctan(y/x) in the first quadrant
(x > 0, y > 0), but possibly requiring an adjustment
for the other quadrants because the signs of both x
and y are individually significant.2
Similarly, it is often necessary to represent basis
vectors of the cylindrical system in terms of Cartesian
basis vectors and vice-versa. Conversion of basis
vectors is straightforward using dot products to
determine the components of the basis vectors in the
new system. For example, x̂ in terms of the basis
vectors of the cylindrical system is
x̂ = ρ̂ (ρ̂ · x̂) + φ̂ φ̂ · x̂ + ẑ (ẑ · x̂)
(4.54)
The last term is of course zero since ẑ · x̂ = 0.
Calculation of the remaining terms requires dot
products between basis vectors in the two systems,
which are summarized in Table 4.1. Using this table,
we find
x̂ = ρ̂ cos φ − φ̂ sin φ
(4.55)
ŷ = ρ̂ sin φ + φ̂ cos φ
(4.56)
and of course ẑ requires no conversion. Going from
Cartesian to cylindrical, we find
ρ̂ = x̂ cos φ + ŷ sin φ
(4.57)
φ̂ = −x̂ sin φ + ŷ cos φ
(4.58)
Integration Over Length. A differential-length
segment of a curve in the cylindrical system is
described in general as
dl = ρ̂dρ + φ̂ρdφ + ẑ dz
(4.59)
Note that the contribution of the φ coordinate to
differential length is ρdφ, not simply dφ. This is
because φ is an angle, not a distance. To see why the
associated distance is ρdφ, consider the following.
2 Note that this function is available in MATLAB and Octave as
atan2(y,x).
4.3. CYLINDRICAL COORDINATES
79
®
²
´dµ
l
ds
°
¯±
d³
x
x
c K. Kikkeri CC BY SA 4.0
c K. Kikkeri CC BY SA 4.0
Figure 4.13: Example in cylindrical coordinates: The
circumference of a circle.
Figure 4.14: Example in cylindrical coordinates: The
area of a circle.
The circumference of a circle of radius ρ is 2πρ. If
only a fraction of the circumference is traversed, the
associated arclength is the circumference scaled by
φ/2π, where φ is the angle formed by the traversed
circumference. Therefore, the distance is
2πρ · φ/2π = ρφ, and the differential distance is ρdφ.
Integration Over Area. Now we ask the question,
what is the integral of some vector field A over a
circular surface S in the z = 0 plane having radius
ρ0 ? This is shown in Figure 4.14. The differential
surface vector in this case is
As always, the integral of a vector field A(r) over a
curve C is
Z
A · dl
(4.60)
The quantities in parentheses are the radial and
angular dimensions, respectively. The direction of ds
indicates the direction of positive flux – see the
discussion in Section 4.2 for an explanation. In
general, the integral over a surface is
Z
A · ds
(4.63)
C
To demonstrate the cylindrical system, let us calculate
the integral of A(r) = φ̂ when C is a circle of radius
ρ0 in the z = 0 plane, as shown in Figure 4.13. In this
example, dl = φ̂ ρ0 dφ since ρ = ρ0 and z = 0 are
both constant along C. Subsequently, A · dl = ρ0 dφ
and the above integral is
Z 2π
ρ0 dφ = 2πρ0
(4.61)
0
i.e., this is a calculation of circumference.
Note that the cylindrical system is an appropriate
choice for the preceding example because the
problem can be expressed with the minimum number
of varying coordinates in the cylindrical system. If we
had attempted this problem in the Cartesian system,
we would find that both x and y vary over C, and in a
relatively complex way.3
3 Nothing will drive this point home more firmly than trying it. It
can be done, but it’s a lot more work...
ds = ẑ (dρ) (ρdφ) = ẑ ρ dρ dφ
(4.62)
S
To demonstrate, let’s consider A = ẑ; in this case
A · ds = ρ dρ dφ and the integral becomes
Z 2π Z ρ0
Z ρ0 Z 2π
dφ
ρ dρ
ρ dρ dφ =
0
0
0
0 1 2
ρ (2π)
=
2 0
= πρ20
(4.64)
which we recognize as the area of the circle, as
expected. The corresponding calculation in the
Cartesian system is quite difficult in comparison.
Whereas the previous example considered a planar
surface, we might consider instead a curved surface.
CHAPTER 4. VECTOR ANALYSIS
80
·
cylinder bounded by ρ ≤ ρ0 and z1 ≤ z ≤ z2 , then
Z
¸0
¹2
dϕ
Z ρ0
0
0
ρ dρ dφ dz
z1
Z 2π Z z2 dφ
dz
ρ dρ
0
= πρ20 (z2 − z1 )
z1
(4.68)
i.e., area times length, which is volume.
º1
Once again, the procedure above is clearly more
complicated than is necessary if we are interested
only in computing volume. However, if the integrand
is not constant-valued then we are no longer simply
computing volume. In this case, the formalism is
appropriate and possibly necessary.
y
¶
c K. Kikkeri CC BY SA 4.0
Additional Reading:
Figure 4.15: Example in cylindrical coordinates: The
area of the curved surface of a cylinder.
Here we go. What is the integral of a vector field
A = ρ̂ over a cylindrical surface S concentric with
the z axis having radius ρ0 and extending from
z = z1 to z = z2 ? This is shown in Figure 4.15. The
differential surface vector in this case is
ds = ρ̂ (ρ0 dφ) (dz) = ρ̂ρ0 dφ dz
(4.65)
The integral is
S
Z ρ0 Z 2π Z z2
0
V
=
dz
Z
A(r) dv =
A · ds =
Z 2π Z z2
0
= ρ0
ρ0 dφdz
z1
Z 2π
0
dφ
Z z2
= 2πρ0 (z2 − z1 )
z1
dz
(4.66)
which is the area of S, as expected. Once again, the
corresponding calculation in the Cartesian system is
quite difficult in comparison.
Integration Over Volume. The differential volume
element in the cylindrical system is
dv = dρ (ρdφ) dz = ρ dρ dφ dz
For example, if A(r) = 1 and the volume V is a
(4.67)
• “Cylindrical coordinate system” on Wikipedia.
4.4. SPHERICAL COORDINATES
81
4.4 Spherical Coordinates
ϕ
[m0097]
The spherical coordinate system is defined with
respect to the Cartesian system in Figure 4.16. The
spherical system uses r, the distance measured from
the origin;4 θ, the angle measured from the +z axis
toward the z = 0 plane; and φ, the angle measured in
a plane of constant z, identical to φ in the cylindrical
system.
Spherical coordinates are preferred over Cartesian
and cylindrical coordinates when the geometry of the
problem exhibits spherical symmetry. For example, in
the Cartesian coordinate system, the surface of a
sphere concentric with the origin requires all three
coordinates (x, y, and z) to describe. However, this
surface can be described using a single constant
parameter – the radius r – in the spherical coordinate
system. This leads to a dramatic simplification in the
mathematics in certain applications.
4 Note that some textbooks use “R” in lieu of r for this coordinate.
z
¼
θ
r
θ
Figure 4.17: Cross products among basis vectors in the
spherical system. (See Figure 4.10 for instructions on
the use of this diagram.)
The basis vectors in the spherical system are r̂, θ̂, and
φ̂. As always, the dot product of like basis vectors is
equal to one, and the dot product of unlike basis
vectors is equal to zero. For the cross-products, we
find:
r̂ × θ̂ = φ̂
(4.69)
θ̂ × φ̂ = r̂
(4.70)
φ̂ × r̂ = θ̂
(4.71)
A useful diagram that summarizes these relationships
is shown in Figure 4.17.
r
θ
r
y
ϕ
»
c K. Kikkeri CC BY SA 4.0
Figure 4.16: Spherical coordinate system and associated basis vectors.
Like the cylindrical system, the spherical system is
often less useful than the Cartesian system for
identifying absolute and relative positions. The
reason is the same: Basis directions in the spherical
system depend on position. For example, r̂ is directed
radially outward from the origin, so r̂ = x̂ for
locations along the x-axis but r̂ = ŷ for locations
along the y axis and r̂ = ẑ for locations along the z
axis. Similarly, the directions of θ̂ and φ̂ vary as a
function of position. To overcome this awkwardness,
it is common to begin a problem in spherical
coordinates, and then to convert to Cartesian
coordinates at some later point in the analysis. Here
are the conversions:
x = r cos φ sin θ
y = r sin φ sin θ
(4.72)
(4.73)
z = r cos θ
(4.74)
The conversion from Cartesian to spherical
CHAPTER 4. VECTOR ANALYSIS
82
·
x̂
ŷ
ẑ
r̂
sin θ cos φ
sin θ sin φ
cos θ
θ̂
cos θ cos φ
cos θ sin φ
− sin θ
z
φ̂
− sin φ
cos φ
0
½¾¿À
dl
θ
a
Table 4.2: Dot products between basis vectors in the
spherical and Cartesian coordinate systems.
coordinates is as follows:
p
r = x2 + y 2 + z 2
θ = arccos (z/r)
φ = arctan (y, x)
(4.75)
(4.76)
(4.77)
(4.78)
where arctan is the four-quadrant inverse tangent
function.5
Dot products between basis vectors in the spherical
and Cartesian systems are summarized in Table 4.2.
This information can be used to convert between basis
vectors in the spherical and Cartesian systems, in the
same manner described in Section 4.3; e.g.
x̂ = r̂ (r̂ · x̂) + θ̂ θ̂ · x̂ + φ̂ φ̂ · x̂
(4.79)
r̂ = x̂ (x̂ · r̂) + ŷ (ŷ · r̂) + ẑ (ẑ · r̂)
ÁÂÃÄ
c K. Kikkeri CC BY SA 4.0
Figure 4.18: Example in spherical coordinates: Poleto-pole distance on a sphere.
Integration Over Length. A differential-length
segment of a curve in the spherical system is
dl = r̂ dr + θ̂ r dθ + φ̂ r sin θ dφ
(4.82)
Note that θ is an angle, as opposed to a distance. The
associated distance is r dθ in the θ direction. Note
also that in the φ direction, distance is r dφ in the
z = 0 plane and less by the factor sin θ for z <> 0.
(4.80)
and so on.
Example 4.3. Cartesian to spherical conversion.
A vector field G = x̂ xz/y. Develop an
expression for G in spherical coordinates.
Solution: Simply substitute expressions in terms
of spherical coordinates for expressions in terms
of Cartesian coordinates. Use Table 4.2 and
Equations 4.72–4.74. Making these substitutions
and applying a bit of mathematical clean-up
afterward, one obtains
G = r̂ sin θ cot φ + φ̂ cos θ cot φ − φ̂
· r cos θ cos φ
ρ
(4.81)
5 Note that this function is available in MATLAB and Octave as
atan2(y,x).
As always, the integral of a vector field A(r) over a
curve C is
Z
C
A · dl
(4.83)
To demonstrate line integration in the spherical
system, imagine a sphere of radius a centered at the
origin with “poles” at z = +a and z = −a. Let us
calculate the integral of A(r) = θ̂, where C is the arc
drawn directly from pole to pole along the surface of
the sphere, as shown in Figure 4.18. In this example,
dl = θ̂ a dθ since r = a and φ (which could be any
value) are both constant along C. Subsequently,
A · dl = a dθ and the above integral is
Z π
a dθ = πa
(4.84)
0
i.e., half the circumference of the sphere, as expected.
Note that the spherical system is an appropriate
choice for this example because the problem can be
4.4. SPHERICAL COORDINATES
83
z
expressed with the minimum number of varying
coordinates in the spherical system. If we had
attempted this problem in the Cartesian system, we
would find that both z and either x or y (or all three)
vary over C and in a relatively complex way.
dϕ
nθ
r si
θ
rd
y
Integration Over Area. Now we ask the question,
what is the integral of some vector field A over the
surface S of a sphere of radius a centered on the
origin? This is shown in Figure 4.19. The differential
surface vector in this case is
a
x
ds = r̂ (r dθ) (r sin θ dφ) = r̂ r2 sin θ dθ dφ (4.85)
c K. Kikkeri CC BY SA 4.0
Figure 4.19: Example in spherical coordinates: The
area of a sphere.
As always, the direction is normal to the surface and
in the direction associated with positive flux. The
quantities in parentheses are the distances associated
with varying θ and φ, respectively. In general, the
integral over a surface is
Z
S
A · ds
sphere of radius a centered on the origin, then
(4.86)
Z
In this case, let’s consider A = r̂; in this case
A · ds = a2 sin θ dθ dφ and the integral becomes
Z π Z 2π
0
0
a2 sin θ dθ dφ = a2
Z π
sin θdθ
0
= a2 (2) (2π)
Z 2π
dφ
0
r2 dr sin θ dθ dφ
Z a
Z π
Z 2π 2
=
r dr
sin θ dθ
dφ
0
0
0 1 3
a (2) (2π)
=
3
4
= πa3
3
(4.89)
A(r) dv =
V
Z a Z π Z 2π
0
0
0
which is the volume of a sphere.
Additional Reading:
= 4πa2
(4.87)
• “Spherical coordinate system” on Wikipedia.
which we recognize as the area of the sphere, as
expected. The corresponding calculation in the
Cartesian or cylindrical systems is quite difficult in
comparison.
Integration Over Volume. The differential volume
element in the spherical system is
dv = dr (rdθ) (r sin θdφ) = r2 dr sin θ dθ dφ
(4.88)
For example, if A(r) = 1 and the volume V is a
CHAPTER 4. VECTOR ANALYSIS
84
4.5 Gradient
In the Cartesian system:
∇f = x̂
[m0098]
The gradient operator is an important and useful tool
in electromagnetic theory. Here’s the main idea:
The gradient of a scalar field is a vector that
points in the direction in which the field is most
rapidly increasing, with the scalar part equal to
the rate of change.
A particularly important application of the gradient is
that it relates the electric field intensity E(r) to the
electric potential field V (r). This is apparent from a
review of Section 2.2 (“Electric Field Intensity”); see
in particular, the battery-charged capacitor example.
In that example, it is demonstrated that:
• The direction of E(r) is the direction in which
V (r) decreases most quickly, and
• The scalar part of E(r) is the rate of change of
V (r) in that direction. Note that this is also
implied by the units, since V (r) has units of V
whereas E(r) has units of V/m.
The gradient is the mathematical operation that
relates the vector field E(r) to the scalar field V (r)
and is indicated by the symbol “∇” as follows:
E(r) = −∇V (r)
(4.90)
or, with the understanding that we are interested in the
gradient as a function of position r, simply
E = −∇V
(4.91)
At this point we should note that the gradient is a very
general concept, and that we have merely identified
one application of the gradient above. In
electromagnetics there are many situations in which
we seek the gradient ∇f of some scalar field f (r).
Furthermore, we find that other differential operators
that are important in electromagnetics can be
interpreted in terms of the gradient operator ∇. These
include divergence (Section 4.6), curl (Section 4.8),
and the Laplacian (Section 4.10).
∂f
∂f
∂f
+ ŷ
+ ẑ
∂x
∂y
∂z
(4.92)
Example 4.4. Gradient of a ramp function.
Find the gradient of f = ax (a “ramp” having
slope a along the x direction).
Solution. Here, ∂f /∂x = a and
∂f /∂y = ∂f /∂z = 0. Therefore ∇f = x̂a.
Note that ∇f points in the direction in which f
most rapidly increases, and has magnitude equal
to the slope of f in that direction.
The gradient operator in the cylindrical and spherical
systems is given in Appendix B.2.
Additional Reading:
• “Gradient” on Wikipedia.
4.6. DIVERGENCE
85
4.6 Divergence
[m0044]
In this section, we present the divergence operator,
which provides a way to calculate the flux associated
with a point in space. First, let us review the concept
of flux.
The integral of a vector field over a surface is a scalar
quantity known as flux. Specifically, the flux F of a
vector field A(r) over a surface S is
Z
A · ds = F
(4.93)
S
Note that A could be fairly described as a flux density;
i.e., a quantity having units equal to the units of F , but
divided by area (i.e., m2 ). Also worth noting is that
the flux of a vector field that has unit magnitude and
is normal to all points on S is simply the area of S.
It is quite useful to identify some electromagnetic
quantities as either fluxes or flux densities. Here are
two important examples:
• The electric flux density D, having units of
C/m2 , is a description of the electric field as a
flux density. (See Section 2.4 for more about
electric flux density.) The integral of D over a
closed surface yields the enclosed charge Qencl ,
having units of C. This relationship is known as
Gauss’ Law:
I
D · ds = Qencl
(4.94)
Summarizing:
Flux is the scalar quantity obtained by integrating
a vector field, interpreted in this case as a flux
density, over a specified surface.
The concept of flux applies to a surface of finite size.
However, what is frequently of interest is behavior at
a single point, as opposed to the sum or average over
a region of space. For example, returning to the idea
of electric flux density (D), perhaps we are not
concerned about the total charge (units of C) enclosed
by a surface, but rather the charge density (C/m3 ) at a
point. In this case, we could begin with Equation 4.94
and divide both sides of the equation by the volume V
enclosed by S:
H
D · ds
Qencl
S
=
(4.96)
V
V
Now we let V shrink to zero, giving us an expression
that must be true at whatever point we decide to
converge upon. Taking the limit as V → 0:
H
D · ds
Qencl
= lim
(4.97)
lim S
V →0 V
V →0
V
The quantity on the right hand side is by definition the
volume charge density ρv (units of C/m3 ) at the point
at which we converge. The left hand side is the
divergence of D, sometimes abbreviated “div D.”
Thus, the above equation can be written
div D = ρv
(4.98)
S
(See Section 5.5 for more about Gauss’ Law.)
• The magnetic flux density B, having units of
Wb/m2 , is a description of the magnetic field as
a flux density. (See Section 2.5 for more about
magnetic flux density.) The integral of B over a
surface (open or closed) yields the magnetic flux
Φ, having units of Wb:
Z
B · ds = Φ
(4.95)
S
This is important because, for example, the time
rate of change of Φ is proportional to electric
potential. (See Section 8.3 for more about this
principle, called Faraday’s Law.)
Summarizing:
Divergence is the flux per unit volume through an
infinitesimally-small closed surface surrounding
a point.
We will typically not actually want to integrate and
take a limit in order to calculate divergence.
Fortunately, we do not have to. It turns out that this
operation can be expressed as the dot product ∇ · D;
where, for example,
∇ , x̂
∂
∂
∂
+ ŷ
+ ẑ
∂x
∂y
∂z
(4.99)
CHAPTER 4. VECTOR ANALYSIS
86
in the Cartesian coordinate system. This is the same
“∇” that appears in the definition of the gradient
operator (Section 4.5) and is same operator that often
arises when considering other differential operators.
If we expand D in terms of its Cartesian components:
D = x̂Dx + ŷDy + ẑDz
(4.100)
Then
∂Dy
∂Dz
∂Dx
+
+
div D = ∇ · D =
∂x
∂y
∂z
To compute divergence in other coordinate systems,
we merely need to know ∇ for those systems. In the
cylindrical system:
∇ = ρ̂
1 ∂
∂
1 ∂
ρ + φ̂
ρ + ẑ
ρ ∂ρ
ρ ∂φ
∂z
(4.102)
and in the spherical system:
(4.101)
This seems to make sense for two reasons. First, it is
dimensionally correct. Taking the derivative of a
quantity having units of C/m2 with respect to distance
yields a quantity having units of C/m3 . Second, it
makes sense that flux from a point should be related
to the sum of the rates of change of the flux density in
each basis direction. Summarizing:
The divergence of a vector field A is ∇ · A.
Example 4.5. Divergence of a uniform field.
A field A that is constant with respect to
position is said to be uniform. A completely
general description of such a field is
A = x̂Ax + ŷAy + ẑAz where Ax , Ay , and Az
are all constants. We see immediately that the
divergence of such a field must be zero. That is,
∇ · A = 0 because each component of A is
constant with respect to position. This also
makes sense from the perspective of the “flux
through an infinitesimally-small closed surface”
interpretation of divergence. If the flux is
uniform, the flux into the surface equals the flux
out of the surface resulting in a net flux of zero.
Example 4.6. Divergence of a
linearly-increasing field.
Consider a field A = x̂A0 x where A0 is a
constant. The divergence of A is ∇ · A = A0 . If
we interpret A as a flux density, then we have
found that the net flux per unit volume is simply
the rate at which the flux density is increasing
with distance.
∂
∂
1 ∂
1
1
∇ = r̂ 2 r2 + θ̂
sin θ + φ̂
r ∂r
r sin θ ∂θ
r sin θ ∂φ
(4.103)
Alternatively, one may use the explicit expressions for
divergence given in Appendix B.2.
Example 4.7. Divergence of a
radially-decreasing field.
Consider a vector field that is directed radially
outward from a point and which decreases
linearly with distance; i.e., A = r̂A0 /r where
A0 is a constant. In this case, the divergence is
most easily computed in the spherical coordinate
system since partial derivatives in all but one
direction (r) equal zero. Neglecting terms that
include these zero-valued partial derivatives, we
find:
A0
A0
1 ∂
r2
= 2
(4.104)
∇·A= 2
r ∂r
r
r
In other words, if we interpret A as a flux
density, then the flux per unit volume is
decreasing with as the square of distance from
the origin.
It is useful to know that divergence, like ∇ itself, is a
linear operator; that is, for any constant scalars a and
b and vector fields A and B:
∇ · (aA + bB) = a∇ · A + b∇ · B
Additional Reading:
• “Divergence” on Wikipedia.
(4.105)
4.7. DIVERGENCE THEOREM
87
4.7 Divergence Theorem
The Divergence Theorem (Equation 4.110) states
that the integral of the divergence of a vector field
over a volume is equal to the flux of that field
through the surface bounding that volume.
[m0046]
The Divergence Theorem relates an integral over a
volume to an integral over the surface bounding that
volume. This is useful in a number of situations that
arise in electromagnetic analysis. In this section, we
derive this theorem.
Consider a vector field A representing a flux density,
such as the electric flux density D or magnetic flux
density B. The divergence of A is
∇·A=f
(4.106)
where is f (r) is the flux per unit volume through an
infinitesimally-small closed surface surrounding the
point at r. Since f is flux per unit volume, we can
obtain flux for any larger contiguous volume V by
integrating over V; i.e.,
Z
flux through V =
f dv
(4.107)
V
In the Cartesian system, V can be interpreted as a
three-dimensional grid of infinitesimally-small cubes
having side lengths dx, dy, and dz, respectively. Note
that the flux out of any face of one of these cubes is
equal to the flux into the cube that is adjacent through
that face. That is, the portion of the total flux that
flows between cubes cancels when added together. In
fact, the only fluxes which do not cancel in the
integration over V are those corresponding to faces
which lie on the bounding surface S, since the
integration stops there. Stating this mathematically:
Z
I
f dv =
A · ds
(4.108)
V
S
Thus, we have converted a volume integral into a
surface integral.
To obtain the Divergence Theorem, we return to
Equation 4.106. Integrating both sides of that
equation over V, we obtain
Z
Z
(∇ · A) dv =
f dv
(4.109)
V
V
Now applying Equation 4.108 to the right hand side:
I
Z
(4.110)
A · ds
(∇ · A) dv =
V
S
The principal utility of the Divergence Theorem is to
convert problems that are defined in terms of
quantities known throughout a volume into problems
that are defined in terms of quantities known over the
bounding surface and vice-versa.
Additional Reading:
• “Divergence theorem” on Wikipedia.
CHAPTER 4. VECTOR ANALYSIS
88
z
4.8 Curl
I
[m0048]
Curl is an operation, which when applied to a vector
field, quantifies the circulation of that field. The
concept of circulation has several applications in
electromagnetics. Two of these applications
correspond to directly to Maxwell’s Equations:
• The circulation of an electric field is proportional
to the rate of change of the magnetic field. This
is a statement of the Maxwell-Faraday Equation
(Section 8.8), which includes as a special case
Kirchoff’s Voltage Law for electrostatics
(Section 5.11).
• The circulation of a magnetic field is
proportional to the source current and the rate of
change of the electric field. This is a statement of
Ampere’s Law (Sections 7.9 and 8.9)
Thus, we are motivated to formally define circulation
and then to figure out how to most conveniently apply
the concept in mathematical analysis. The result is the
curl operator.
So, we begin with the concept of circulation:
“Circulation” is the integral of a vector field over
a closed path.
Specifically, the circulation of the vector field A(r)
over the closed path C is
I
A · dl
(4.111)
C
The circulation of a uniform vector field is zero for
any valid path. For example, the circulation of
A = x̂A0 is zero because non-zero contributions at
each point on C cancel out when summed over the
closed path. On the other hand, the circulation of
A = φ̂A0 over a circular path of constant ρ and z is a
non-zero constant, since the non-zero contributions to
the integral at each point on the curve are equal and
accumulate when summed over the path.
H
c K. Kikkeri CC BY SA 4.0
Figure 4.20: Magnetic field intensity due to a current
flowing along the z axis.
Example 4.8. Circulation of the magnetic field
intensity surrounding a line current.
Consider a current I (units of A) flowing along
the z axis in the +z direction, as shown in
Figure 4.20. It is known that this current gives
rise to a magnetic field intensity H = φ̂H0 /ρ,
where H0 is a constant having units of A since
the units of H are A/m. (Feel free to consult
Section 7.5 for the details; however, no
additional information is needed to follow the
example being presented here.) The circulation
of H along any circular path of radius a in a
plane of constant z is therefore
Z 2π I
H0
· φ̂ a dφ = 2πH0
φ̂
H · dl =
a
φ=0
C
Note that the circulation of H in this case has
two remarkable features: (1) It is independent of
the radius of the path of integration; and (2) it
has units of A, which suggests a current. In fact,
it turns out that the circulation of H in this case
is equal to the enclosed source current I.
Furthermore, it turns out that the circulation of
H along any path enclosing the source current is
equal to the source current! These findings are
consequences of Ampere’s Law, as noted above.
Curl is, in part, an answer to the question of what the
circulation at a point in space is. In other words, what
4.8. CURL
is the circulation as C shrinks to it’s smallest possible
size. The answer in one sense is zero, since the
arclength of C is zero in this limit – there is nothing to
integrate over. However, if we ask instead what is the
circulation per unit area in the limit, then the result
should be the non-trivial value of interest. To express
this mathematically, we constrain C to lie in a plane,
and define S to be the open surface bounded by C in
this case. Then, we define the scalar part of the curl of
A to be:
H
A · dl
(4.112)
lim C
∆s→0
∆s
where ∆s is the area of S, and (important!) we
require C and S to lie in the plane that maximizes the
above result.
Because S and it’s boundary C lie in a plane, it is
possible to assign a direction to the result. The chosen
direction is the normal n̂ to the plane in which C and
S lie. Because there are two normals at each point on
a plane, we specify the one that satisfies the right
hand rule. This rule, applied to the curl, states that the
correct normal is the one which points through the
plane in the same direction as the fingers of the right
hand when the thumb of your right hand is aligned
along C in the direction of integration. Why is this the
correct orientation of n̂? See Section 4.9 for the
answer to that question. For the purposes of this
section, it suffices to consider this to be simply an
arbitrary sign convention.
Now with the normal vector n̂ unambiguously
defined, we can now formally define the curl
operation as follows:
H
n̂ C A · dl
curl A , lim
(4.113)
∆s→0
∆s
where, once again, ∆s is the area of S, and we select
S to lie in the plane that maximizes the magnitude of
the above result. Summarizing:
The curl operator quantifies the circulation of a
vector field at a point.
The magnitude of the curl of a vector field is the
circulation, per unit area, at a point and such that
the closed path of integration shrinks to enclose
zero area while being constrained to lie in the
plane that maximizes the magnitude of the result.
89
The direction of the curl is determined by the
right-hand rule applied to the associated path of
integration.
Curl is a very important operator in electromagnetic
analysis. However, the definition (Equation 4.113) is
usually quite difficult to apply. Remarkably, however,
it turns out that the curl operation can be defined in
terms of the ∇ operator; that is, the same ∇ operator
associated with the gradient, divergence, and
Laplacian operators. Here is that definition:
curl A , ∇ × A
(4.114)
For example: In Cartesian coordinates,
∂
∂
∂
+ ŷ
+ ẑ
∂x
∂y
∂z
(4.115)
A = x̂Ax + ŷAy + ẑAz
(4.116)
∇ , x̂
and
so curl can be calculated as follows:
∇×A=
x̂
ŷ
ẑ
∂
∂x
∂
∂y
∂
∂z
Ax
Ay
Az
or, evaluating the determinant:
∂Az
∂Ay
∇ × A = x̂
−
∂y
∂z
∂Ax
∂Az
+ ŷ
−
∂z
∂x
∂Ay
∂Ax
+ ẑ
−
∂x
∂y
(4.117)
(4.118)
Expressions for curl in each of the three major
coordinate systems is provided in Appendix B.2.
It is useful to know is that curl, like ∇ itself, is a
linear operator; that is, for any constant scalars a and
b and vector fields A and B:
∇ × (aA + bB) = a∇ × A + b∇ × B
Additional Reading:
• “Curl (mathematics)” on Wikipedia.
(4.119)
CHAPTER 4. VECTOR ANALYSIS
90
4.9 Stokes’ Theorem
Additional Reading:
• “Stokes’ Theorem” on Wikipedia.
[m0051]
Stokes’ Theorem relates an integral over an open
surface to an integral over the curve bounding that
surface. This relationship has a number of
applications in electromagnetic theory. Here is the
theorem:
Z
S
(∇ × A) · ds =
I
C
A · dl
(4.120)
where S is the open surface bounded by the closed
path C. The direction of the surface normal ds = n̂ds
is related to the direction of integration along C by the
right-hand rule, illustrated in Figure 4.21. In this
case, the right-hand rule states that the correct normal
is the one that points through the surface in the same
direction as the fingers of the right hand when the
thumb of your right hand is aligned along C in the
direction of integration.
Stokes’ Theorem is a purely mathematical result and
not a principle of electromagnetics per se. The
relevance of the theorem to electromagnetic theory is
primarily as a tool in the associated mathematical
analysis. Usually the theorem is employed to
transform a problem expressed in terms of an
integration over a surface into an integration over a
closed path or vice-versa. For more information on
the theorem and its derivation, see “Additional
Reading” at the end of this section.
c Cronholm144 (modified) CC BY SA 3.0
Figure 4.21: The relative orientations of the direction of integration C and surface normal n̂ in Stokes’
Theorem.
4.10. THE LAPLACIAN OPERATOR
91
4.10 The Laplacian Operator
It is sometimes useful to know that the Laplacian of a
vector field can be expressed in terms of the gradient,
divergence, and curl as follows:
[m0099]
∇2 A = ∇ (∇ · A) − ∇ × (∇ × A)
The Laplacian ∇2 f of a field f (r) is the divergence
of the gradient of that field:
∇2 f , ∇ · (∇f )
(4.121)
Note that the Laplacian is essentially a definition of
the second derivative with respect to the three spatial
dimensions. For example, in Cartesian coordinates,
∇2 f =
∂2f
∂2f
∂2f
+ 2 + 2
2
∂x
∂y
∂z
(4.122)
The Laplacian operator in the cylindrical and
spherical coordinate systems is given in
Appendix B.2.
Additional Reading:
• “Laplace operator” on Wikipedia.
[m0043]
as can be readily verified by applying the definitions
of gradient and divergence in Cartesian coordinates to
Equation 4.121.
The Laplacian relates the electric potential (i.e., V ,
units of V) to electric charge density (i.e., ρv , units of
C/m3 ). This relationship is known as Poisson’s
Equation (Section 5.15):
∇2 V = −
ρv
ǫ
(4.123)
where ǫ is the permittivity of the medium. The fact
that V is related to ρv in this way should not be
surprising, since electric field intensity (E, units of
V/m) is proportional to the derivative of V with
respect to distance (via the gradient) and ρv is
proportional to the derivative of E with respect to
distance (via the divergence).
The Laplacian operator can also be applied to vector
fields; for example, Equation 4.122 is valid even if the
scalar field “f ” is replaced with a vector field. In the
Cartesian coordinate system, the Laplacian of the
vector field A = x̂Ax + ŷAy + ẑAz is
∇2 A = x̂∇2 Ax + ŷ∇2 Ay + ẑ∇2 Az
(4.124)
An important application of the Laplacian operator of
vector fields is the wave equation; e.g., the wave
equation for E in a lossless and source-free region is
(Section 9.2)
∇2 E + β 2 E = 0
(4.125)
where β is the phase propagation constant.
(4.126)
92
CHAPTER 4. VECTOR ANALYSIS
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 4.7: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0006 fDotProductSpecialCases.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 4.9: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0006 fCrossProduct.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 4.11: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0096 fCylindricalCoordinates.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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https://commons.wikimedia.org/wiki/File:Cylindrical Coordinates-Area of a Circle.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 4.16: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:Spherical Coordinate System.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
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CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Heavily modified by author.
Chapter 5
Electrostatics
[m0116]
5.1 Coulomb’s Law
Electrostatics is the theory of the electric field in
conditions in which its behavior is independent of
magnetic fields, including
[m0102]
• The electric field associated with fixed
distributions of electric charge
• Capacitance (the ability of a structure to store
energy in an electric field)
• The energy associated with the electrostatic field
• Steady current induced in a conducting material
in the presence of an electrostatic field
(essentially, Ohm’s Law)
Consider two charge-bearing particles in free space,
identified as “particle 1” and “particle 2” in
Figure 5.1. Let the charges borne by these particles be
q1 and q2 , and let R be the distance between them. If
the particles bear charges of the same sign (i.e., if
q1 q2 is positive), then the particles repel; otherwise,
they attract. This repulsion or attraction can be
quantified as a force experienced by each particle.
Physical observations reveal that the magnitude of the
force is proportional to q1 q2 , and inversely
proportional to R2 . For particle 2 we find:
F = R̂F0
q1 q2
R2
(5.1)
The term “static” refers to the fact that these aspects
of electromagnetic theory can be developed by
assuming sources are time-invariant; we might say
that electrostatics is the study of the electric field at
DC. However, many aspects of electrostatics are
relevant to AC, radio frequency, and higher-frequency
applications as well.
where R̂ is the unit vector pointing from the particle 1
to the particle 2, and F0 is a constant. The value of F0
must have units of inverse permittivity; i.e., (F/m)−1 .
This is most easily seen by dimensional analysis of
the above relationship, including the suspected factor:
Additional Reading:
where we have used the facts that 1 F = 1 C/V, 1 V =
1 J/C, and 1 N = 1 J/m. This finding suggests that
F0 ∝ ǫ−1 , where ǫ is the permittivity of the medium
in which the particles exist. Observations confirm that
the force is in fact inversely proportional to the
permittivity, with an additional factor of 1/4π
(unitless). Putting this all together we obtain what is
commonly known as Coulomb’s Law:
• “Electrostatics” on Wikipedia.
C·C
C·C
C·C
C·V
J
=
=
=
=
=N
F/m · m2
F·m
C/V · m
m
m
F = R̂
q1 q2
4πǫR2
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
(5.2)
CHAPTER 5. ELECTROSTATICS
94
F
q2
Example 5.1. Electric field of a point charge at
the origin.
particle
q1
Å
R
A common starting point in electrostatic analysis
is the field associated with a particle bearing
charge q at the origin of the coordinate system.
Because the electric field is directed radially
away from a positively-charged source particle
in all directions, this field is most conveniently
described in the spherical coordinate system.
Thus, R̂ becomes r̂, R becomes r, and we have
particle
ÇF
Æ
c K. Kikkeri CC BY SA 4.0
Figure 5.1: Coulomb’s Law describes the force perceived by pairs of charged particles.
Subsequently, the force perceived by particle 2 is
equal and opposite; i.e., equal to −F.
Separately, it is known that F can be described in
terms of the electric field intensity E1 associated with
particle 1:
F = q2 E1
(5.3)
This is essentially the definition of E1 , as explained in
Section 2.2. Combining this result with Coulomb’s
Law, we obtain a means to directly calculate the field
associated with the first particle in the absence of the
second particle:
E1 = R̂
q1
4πǫR2
(5.4)
where now R̂R is the vector beginning at the
particle 1 and ending at the point to be evaluated.
The electric field intensity associated with a point
charge (Equation 5.4) is (1) directed away from
positive charge, (2) proportional to the magnitude of the charge, (3) inversely proportional to
the permittivity of the medium, and (3) inversely
proportional to distance squared.
We have described this result as originating from
Coulomb’s Law, which is based on physical
observations. However, the same result may be
obtained directly from Maxwell’s Equations using
Gauss’ Law (Section 5.5).
E(r) = r̂
q
4πǫr2
(5.5)
Here’s a numerical example. What is the electric
field intensity at a distance 1 µm from a single
electron located at the origin, in free space? In
this case, q ∼
= −1.60 × 10−19 C (don’t forget
that minus sign!), ǫ = ǫ0 , r = 1 µm, and we
find:
E(r) = −r̂ (1.44 kV/m)
(5.6)
This is large relative to electric field strengths
commonly encountered in engineering
applications. The strong electric field of the
electron is not readily apparent because electrons
in common materials tend to be accompanied by
roughly equal amounts of positive charge, such
as the protons of atoms. Sometimes, however,
the effect of individual electrons does become
significant in practical electronics through a
phenomenon known as shot noise.
Additional Reading:
• “Coulomb’s Law” on Wikipedia.
• “Shot Noise” on Wikipedia.
5.2. ELECTRIC FIELD DUE TO POINT CHARGES
5.2 Electric Field Due to Point
Charges
95
5.3 Charge Distributions
[m0100]
[m0103]
The electric field intensity associated with a single
particle bearing charge q1 , located at the origin, is
(Section 5.1)
q1
(5.7)
E(r) = r̂
4πǫr2
If this particle is instead located at some position r1 ,
then the above expression may be written as follows:
E(r; r1 ) =
r − r1
q1
|r − r1 | 4πǫ |r − r1 |2
(5.8)
or, combining like terms in the denominator:
E(r; r1 ) =
r − r 1 q1
3
|r − r1 | 4πǫ
(5.9)
Now let us consider the field due to multiple such
particles. Under the usual assumptions about the
permittivity of the medium (reminder: Section 2.8),
the property of superposition applies. Using this
principle, we conclude:
The electric field resulting from a set of charged
particles is equal to the sum of the fields associated with the individual particles.
Stated mathematically:
E(r) =
N
X
E(r; rn )
(5.10)
In principle, the smallest unit of electric charge that
can be isolated is the charge of a single electron,
which is ∼
= −1.60 × 10−19 C. This is very small, and
we rarely deal with electrons one at a time, so it is
usually more convenient to describe charge as a
quantity that is continuous over some region of space.
In particular, it is convenient to describe charge as
being distributed in one of three ways: along a curve,
over a surface, or within a volume.
Line Charge Distribution. Imagine that charge is
distributed along a curve C through space. Let ∆q be
the total charge along a short segment of the curve,
and let ∆l be the length of this segment. The line
charge density ρl at any point along the curve is
defined as
dq
∆q
ρl , lim
=
(5.12)
∆l→0 ∆l
dl
which has units of C/m. We may then define ρl to be
a function of position along the curve, parameterized
by l; e.g., ρl (l). Then, the total charge Q along the
curve is
Z
Q=
ρl (l) dl
(5.13)
C
which has units of C. In other words, line charge
density integrated over length yields total charge.
Surface Charge Distribution. Imagine that charge is
distributed over a surface. Let ∆q be the total charge
on a small patch on this surface, and let ∆s be the
area of this patch. The surface charge density ρs at
any point on the surface is defined as
n=1
where N is the number of particles. Thus, we have
N
1 X r − rn
E(r) =
qn
4πǫ n=1 |r − rn |3
(5.11)
∆q
ρs , lim
∆s→0 ∆s
=
dq
ds
(5.14)
which has units of C/m2 . Let us define ρs to be a
function of position on this surface. Then the total
charge over a surface S is
Z
Q=
ρs ds
(5.15)
S
In other words, surface charge density integrated over
a surface yields total charge.
Volume Charge Distribution. Imagine that charge is
distributed over a volume. Let ∆q be the total charge
CHAPTER 5. ELECTROSTATICS
96
in a small cell within this volume, and let ∆v be the
volume of this cell. The volume charge density ρv at
any point in the volume is defined as
∆q
ρv , lim
∆v→0 ∆v
=
dq
dv
5.4 Electric Field Due to a
Continuous Distribution of
Charge
(5.16)
which has units of C/m3 . Since ρv is a function of
position within this volume, the total charge within a
volume V is
Z
Q=
ρv dv
(5.17)
V
In other words, volume charge density integrated over
a volume yields total charge.
[m0104]
The electric field intensity associated with N charged
particles is (Section 5.2):
N
E(r) =
1 X r − rn
qn
4πǫ n=1 |r − rn |3
(5.18)
where qn and rn are the charge and position of the
nth particle. However, it is common to have a
continuous distribution of charge as opposed to a
countable number of charged particles. In this
section, we extend Equation 5.18 using the concept of
continuous distribution of charge (Section 5.3) so that
we may address this more general class of problems.
Distribution of Charge Along a Curve. Consider a
continuous distribution of charge along a curve C.
The curve can be divided into short segments of
length ∆l. Then, the charge associated with the nth
segment, located at rn , is
qn = ρl (rn ) ∆l
(5.19)
where ρl is charge density (units of C/m) at rn .
Substituting this expression into Equation 5.18, we
obtain
N
1 X r − rn
ρl (rn ) ∆l
E(r) =
4πǫ n=1 |r − rn |3
(5.20)
Taking the limit as ∆l → 0 yields:
E(r) =
1
4πǫ
Z
r − r′
′ 3
C |r − r |
ρl (r′ ) dl
(5.21)
where r′ represents the varying position along C with
integration.
The simplest example of a curve is a straight line. It is
straightforward to use Equation 5.21 to determine the
electric field due to a distribution of charge along a
straight line. However, it is much easier to analyze
that particular distribution using Gauss’ Law, as
5.4. ELECTRIC FIELD DUE TO A CONTINUOUS DISTRIBUTION OF CHARGE
97
z
shown in Section 5.6. The following example
addresses a charge distribution for which
Equation 5.21 is more appropriate.
ÈÉÊË
point
y
r - rÌ
Example 5.2. Electric field along the axis of a
ring of uniformly-distributed charge.
Consider a ring of radius a in the z = 0 plane,
centered on the origin, as shown in Figure 5.2.
Let the charge density along this ring be uniform
and equal to ρl (C/m). Find the electric field
along the z axis.
ρa
Solution. The source charge position is given in
cylindrical coordinates as
r′ = ρ̂a
(5.22)
The position of a field point along the z axis is
simply
r = ẑz
(5.23)
c K. Kikkeri CC BY SA 4.0
Figure 5.2: Calculating the electric field along the
axis of a ring of charge.
we obtain
E(z) = ẑ
Thus,
′
r − r = −ρ̂a + ẑz
(5.24)
p
a2 + z 2
(5.25)
and
|r − r′ | =
Equation 5.21 becomes:
E(z) =
x
1
4πǫ
Z 2π
0
−ρ̂a + ẑz
[a2 + z 2 ]
3/2
z
ρl a
2ǫ [a2 + z 2 ]3/2
(5.28)
It is a good exercise to confirm that this result is
dimensionally correct. It is also recommended to
confirm that when z ≫ a, the result is
approximately the same as that expected from a
particle having the same total charge as the ring.
ρl (a dφ)
(5.26)
Pulling factors that do not vary with φ out of the
integral and factoring into separate integrals for
the φ̂ and ẑ components, we obtain:
Z 2π
Z 2π ρl a
dφ
ρ̂
dφ
+
ẑz
−a
3/2
0
0
4πǫ [a2 + z 2 ]
(5.27)
The second integral is equal to 2π. The first
integral is equal to zero. To see this, note that the
integral is simply summing values of ρ̂ for all
possible values of φ. Since ρ̂(φ + π) = −ρ̂(φ),
the integrand for any given value of φ is equal
and opposite the integrand π radians later. (This
is one example of a symmetry argument.) Thus,
Distribution of Charge Over a Surface. Consider a
continuous distribution of charge over a surface S.
The surface can be divided into small patches having
area ∆s. Then, the charge associated with the nth
patch, located at rn , is
qn = ρs (rn ) ∆s
(5.29)
where ρs is the surface charge density (units of C/m2 )
at rn . Substituting this expression into Equation 5.18,
we obtain
N
E(r) =
1 X r − rn
ρs (rn ) ∆s
4πǫ n=1 |r − rn |3
(5.30)
Taking the limit as ∆s → 0 yields:
E(r) =
1
4πǫ
Z
r − r′
′
S |r − r |
′
3 ρs (r ) ds
(5.31)
CHAPTER 5. ELECTROSTATICS
98
where r′ represents the varying position over S with
integration.
Example 5.3. Electric field along the axis of a
disk of uniformly-distributed charge.
Consider a circular disk of radius a in the z = 0
plane, centered on the origin, as shown in
Figure 5.3. Let the charge density over this disk
be uniform and equal to ρs (C/m2 ). Find the
electric field along the z axis.
= ẑ2πz
(5.32)
The position of a field point along the z axis is
simply
r = ẑz
(5.33)
Thus,
r − r′ = −ρ̂ρ + ẑz
and
(5.34)
p
|r − r′ | = ρ2 + z 2
(5.35)
1
4πǫ
Z a Z 2π
ρ=0
−ρ̂ρ + ẑz
φ=0 [ρ2 + z 2 ]
3/2
ρs (ρ dρ dφ)
(5.36)
To solve this integral, first rearrange the double
integral into a single integral over φ followed by
integration over ρ:
Z 2π
Z a
ρ
ρs
(−ρ̂ρ
+
ẑz)
dφ
dρ
4πǫ ρ=0 [ρ2 + z 2 ]3/2 φ=0
(5.37)
Now we address the integration over φ shown in
the square brackets in the above expression:
Z 2π
φ=0
(−ρ̂ρ + ẑz) dφ = −ρ
Z 2π
φ=0
(5.39)
Substituting this into Expression 5.37 we obtain:
Z a
ρs
ρ
[ẑ2πz] dρ
4πǫ ρ=0 [ρ2 + z 2 ]3/2
Z
ρs z a
ρ dρ
=ẑ
(5.40)
2ǫ ρ=0 [ρ2 + z 2 ]3/2
This integral can be solved using integration by
parts and trigonometric substitution. Since the
solution is tedious and there is no particular
principle of electromagnetics demonstrated by
this solution, we shall simply state the result:
Z a
ρ dρ
3/2
ρ=0 [ρ2 + z 2 ]
= p
−1
a
ρ2 + z 2 ρ=0
1
−1
(5.41)
+
=√
2
2
|z|
a +z
Equation 5.31 becomes:
E(z) =
φ=0
φ=0
Solution. The source charge position is given in
cylindrical coordinates as
r′ = ρ̂ρ
the contribution from each pointing of ρ̂ is
canceled out by another pointing of ρ̂ that is in
the opposite direction. Since there is an equal
number of these canceling pairs of pointings, the
result is zero. Thus:
Z 2π
Z 2π
dφ
(−ρ̂ρ + ẑz) dφ = 0 + ẑz
ρ̂dφ+ẑz
Z 2π
dφ
φ=0
(5.38)
The first integral on the right is zero for the
following reason. As the integral progresses in
φ, the vector ρ̂ rotates. Because the integration is
over a complete revolution (i.e., φ from 0 to 2π),
Substituting this result:
1
ρs z
−1
√
+
E(z) = ẑ
2ǫ
|z|
a2 + z 2
z
ρs
−z
√
+
= ẑ
2ǫ
|z|
a2 + z 2
ρs
−z
√
+ sgn z
= ẑ
2ǫ
a2 + z 2
(5.42)
where “sgn” is the “signum” function; i.e.,
sgn z = +1 for z > 0 and sgn z = −1 for
z < 0. Summarizing:
ρs
z
E(z) = ẑ
sgn z − √
(5.43)
2ǫ
a2 + z 2
It is a good exercise to confirm that this result is
dimensionally correct and yields an electric field
vector that points in the expected direction and
with the expected dependence on a and z.
5.4. ELECTRIC FIELD DUE TO A CONTINUOUS DISTRIBUTION OF CHARGE
z
Taking the limit as ∆v → 0 yields:
ÎÏÐÑ
E(r) =
y
point
r - rÍ
1
4πǫ
Z
r − r′
V
3
|r − r′ |
ρv (r′ ) dv
(5.47)
where r′ represents the varying position over V with
integration.
ρρ
x
a
c K. Kikkeri CC BY SA 4.0
Figure 5.3: Calculating the electric field along the
axis of a disk of charge.
A special case of the “disk of charge” scenario
considered in the preceding example is an infinite
sheet of charge. The electric field from an infinite
sheet of charge is a useful theoretical result. We get
the field in this case simply by letting a → ∞ in
Equation 5.43, yielding:
E(r) = ẑ
ρs
sgn z
2ǫ
(5.44)
Again, it is useful to confirm that this is
dimensionally correct: C/m2 divided by F/m yields
V/m. Also, note that Equation 5.44 is the electric field
at any point above or below the charge sheet – not just
on z axis. This follows from symmetry. From the
perspective of any point in space, the edges of the
sheet are the same distance (i.e., infinitely far) away.
Distribution of Charge In a Volume. Consider a
continuous distribution of charge within a volume V.
The volume can be divided into small cells (volume
elements) having volume ∆v. Then, the charge
associated with the nth cell, located at rn , is
qn = ρv (rn ) ∆v
(5.45)
where ρv is volume charge density (units of C/m3 ) at
rn . Substituting this expression into Equation 5.18,
we obtain
N
E(r) =
99
1 X r − rn
ρv (rn ) ∆v
4πǫ n=1 |r − rn |3
(5.46)
CHAPTER 5. ELECTROSTATICS
100
5.5 Gauss’ Law: Integral Form
[m0014]
Gauss’ Law is one of the four fundamental laws of
classical electromagnetics, collectively known as
Maxwell’s Equations. Before diving in, the reader is
strongly encouraged to review Section 2.4. In that
section, Gauss’ Law emerges from the interpretation
of the electric field as a flux density. Section 2.4 does
not actually identify Gauss’ Law, but here it is:
Gauss’ Law (Equation 5.48) states that the flux of
the electric field through a closed surface is equal
to the enclosed charge.
Gauss’ Law is expressed mathematically as follows:
I
(5.48)
D · ds = Qencl
S
where D is the electric flux density ǫE, S is a closed
surface with differential surface normal ds, and Qencl
is the enclosed charge. We can see the law is
dimensionally correct; D has units of C/m2 , thus
integrating D over a surface gives a quantity with
units of C/m2 · m2 = C, which are the units of charge.
Gauss’ Law has a number of applications in
electromagnetic theory. One of them, as explored
below, is as a method to compute the electric field in
response to a distribution of electric charge. Note that
a method to do this, based on Coulomb’s Law, is
described in Sections 5.1, 5.2, and 5.4. Gauss’ Law
provides an alternative method that is easier or more
useful in certain applications.
Example 5.4. Electric field associated with a
charged particle, using Gauss’ Law.
In this example, we demonstrate the ability of
Gauss’ Law to predict the field associated with a
charge distribution. Let us do this for the
simplest possible charge distribution. A particle
of charge q located at the origin, for which we
already have the answer (Section 5.1).
Solution. Gauss’ Law applies to any surface that
encloses the charge, so for simplicity we chose a
sphere of radius r centered at the origin. Note
that Qencl on the right hand side is just q for any
surface having r > 0. Gauss’ Law in this case
becomes
Z π Z 2π
D · r̂r2 sin θ dθ dφ = q (5.49)
θ=0
φ=0
If we can solve for D, we can get E using
D = ǫE. The simplest way to solve for D is to
use a symmetry argument, which proceeds as
follows. In this problem, the magnitude of D
can depend only on r, and not θ or φ. This is
because the charge has no particular orientation,
and the sphere is centered on the charge.
Similarly, it is clear that D must point either
directly toward or directly away from the charge.
In other words, D = r̂D(r). Substituting this in
the above equation, we encounter the dot
product r̂ · r̂, which is simply 1. Since D(r) and
r2 are constants with respect to the integration,
we obtain:
Z π Z 2π
sin θ dθdφ = q
(5.50)
r2 D(r)
θ=0
φ=0
The remaining integral is simply 4π, thus we
obtain:
q
(5.51)
D(r) =
4πr2
Bringing the known vector orientation of D
back into the equation, we obtain
D = r̂
q
4πr2
(5.52)
and finally using D = ǫE we obtain the
expected result
E = r̂
q
4πǫr2
(5.53)
Here’s the point you should take away from the above
example:
Gauss’ Law combined with a symmetry argument may be sufficient to determine the electric
field due to a charge distribution. Thus, Gauss’
Law may be an easier alternative to Coulomb’s
Law in some applications.
5.6. ELECTRIC FIELD DUE TO AN INFINITE LINE CHARGE USING GAUSS’ LAW
Additional Reading:
• “Gauss’ Law” on Wikipedia.
101
5.6 Electric Field Due to an
Infinite Line Charge using
Gauss’ Law
[m0149]
Section 5.5 explains one application of Gauss’ Law,
which is to find the electric field due to a charged
particle. In this section, we present another
application – the electric field due to an infinite line of
charge. The result serves as a useful “building block”
in a number of other problems, including
determination of the capacitance of coaxial cable
(Section 5.24). Although this problem can be solved
using the “direct” approach described in Section 5.4
(and it is an excellent exercise to do so), the Gauss’
Law approach demonstrated here turns out to be
relatively simple.
Example 5.5. Electric field associated with an
infinite line charge, using Gauss’ Law.
Use Gauss’ Law to determine the electric field
intensity due to an infinite line of charge along
the z axis, having charge density ρl (units of
C/m), as shown in Figure 5.4.
Solution. Gauss’ Law requires integration over
a surface that encloses the charge. So, our first
problem is to determine a suitable surface. A
cylinder of radius a that is concentric with the z
axis, as shown in Figure 5.4, is maximally
symmetric with the charge distribution and so is
likely to yield the simplest possible analysis. At
first glance, it seems that we may have a
problem since the charge extends to infinity in
the +z and −z directions, so it’s not clear how
to enclose all of the charge. Let’s suppress that
concern for a moment and simply choose a
cylinder of finite length l. In principle, we can
solve the problem first for this cylinder of finite
size, which contains only a fraction of the
charge, and then later let l → ∞ to capture the
rest of the charge. (In fact, we’ll find when the
time comes it will not be necessary to do that,
but we shall prepare for it anyway.)
CHAPTER 5. ELECTROSTATICS
102
Here’s Gauss’ Law:
I
D · ds = Qencl
(5.54)
S
where D is the electric flux density ǫE, S is a
closed surface with outward-facing differential
surface normal ds, and Qencl is the enclosed
charge.
The first order of business is to constrain the
form of D using a symmetry argument, as
follows. Consider the field of a point charge q at
the origin (Section 5.5):
D = r̂
q
4πr2
(5.55)
We can “assemble” an infinite line of charge by
adding particles in pairs. One pair is added at a
time, with one particle on the +z axis and the
other on the −z axis, with each located an equal
distance from the origin. We continue to add
particle pairs in this manner until the resulting
charge extends continuously to infinity in both
directions. The principle of superposition
indicates that the resulting field will be the sum
of the fields of the particles (Section 5.2). Thus,
we see that D cannot have any component in the
φ̂ direction because none of the fields of the
constituent particles have a component in that
direction. Similarly, we see that the magnitude
of D cannot depend on φ because none of the
fields of the constituent particles depends on φ
and because the charge distribution is identical
(“invariant”) with rotation in φ. Also, note that
for any choice of z the distribution of charge
above and below that plane of constant z is
identical; therefore, D cannot be a function of z
and D cannot have any component in the ẑ
direction. Therefore, the direction of D must be
radially outward; i.e., in the ρ̂ direction, as
follows:
D = ρ̂Dρ (ρ)
(5.56)
Next, we observe that Qencl on the right hand
side of Equation 5.54 is equal to ρl l. Thus, we
obtain
I
[ρ̂Dρ (ρ)] · ds = ρl l
(5.57)
S
The cylinder S consists of a flat top, curved side,
and flat bottom. Expanding the above equation
to reflect this, we obtain
Z
ρl l =
[ρ̂Dρ (ρ)] · (+ẑds)
top
Z
+
[ρ̂Dρ (ρ)] · (+ρ̂ds)
Zside
+
[ρ̂Dρ (ρ)] · (−ẑds)
(5.58)
bottom
Examination of the dot products indicates that
the integrals associated with the top and bottom
surfaces must be zero. In other words, the flux
through the top and bottom is zero because D is
perpendicular to these surfaces. We are left with
Z
ρl l =
[Dρ (ρ)] ds
(5.59)
side
The side surface is an open cylinder of radius
ρ = a, so Dρ (ρ) = Dρ (a), a constant over this
surface. Thus:
Z
Z
ρl l =
[Dρ (a)] ds = [Dρ (a)]
ds
side
side
(5.60)
The remaining integral is simply the area of the
side surface, which is 2πa · l. Solving for Dρ (a)
we obtain
Dρ (a) =
ρl
ρl l
=
2πal
2πa
(5.61)
Remarkably, we see Dρ (a) is independent of l,
So the concern raised in the beginning of this
solution – that we wouldn’t be able to enclose all
of the charge – doesn’t matter.
Completing the solution, we note the result must
be the same for any value of ρ (not just ρ = a),
so
ρl
D = ρ̂Dρ (ρ) = ρ̂
(5.62)
2πρ
and since D = ǫE:
E = ρ̂
ρl
2πǫρ
(5.63)
This completes the solution. We have found that
the electric field is directed radially away from
5.7. GAUSS’ LAW: DIFFERENTIAL FORM
z
103
5.7 Gauss’ Law: Differential
Form
Ôρ
Ò l
Ó
[m0045]
ρ
a
c K. Kikkeri CC BY SA 4.0
Figure 5.4: Finding the electric field of an infinite line
of charge using Gauss’ Law.
the line charge, and decreases in magnitude in
inverse proportion to distance from the line
charge. Suggestion: Check to ensure that this
solution is dimensionally correct.
The integral form of Gauss’ Law (Section 5.5) is a
calculation of enclosed charge Qencl using the
surrounding density of electric flux:
I
D · ds = Qencl
(5.64)
S
where D is electric flux density and S is the enclosing
surface. It is also sometimes necessary to do the
inverse calculation (i.e., determine electric field
associated with a charge distribution). This is
sometimes possible using Equation 5.64 if the
symmetry of the problem permits; see examples in
Section 5.5 and 5.6. If the problem does not exhibit
the necessary symmetry, then it seems that one must
fall back to the family of techniques presented in
Section 5.4 requiring direct integration over the
charge, which is derived from Coulomb’s Law.
However, even the Coulomb’s Law / direct integration
approach has a limitation that is very important to
recognize: It does not account for the presence of
structures that may influence the electric field. For
example, the electric field due to a charge in free
space is different from the electric field due to the
same charge located near a perfectly-conducting
surface. In fact, these approaches do not account for
the possibility of any spatial variation in material
composition, which rules out their use in many
engineering applications.
To address this broader scope of problems, we require
an alternative form of Gauss’ Law that applies at
individual points in space. That is, we require Gauss’
Law expressed in the form of a differential equation,
as opposed to an integral equation. This facilitates the
use of Gauss’ Law even in problems that do not
exhibit sufficient symmetry and that involve material
boundaries and spatial variations in material
constitutive parameters. Given this differential
equation and the boundary conditions imposed by
structure and materials, we may then solve for the
electric field in these more complicated scenarios. In
this section, we derive the desired differential form of
Gauss’ Law. Elsewhere (in particular, in Section 5.15)
CHAPTER 5. ELECTROSTATICS
104
we use this equation as a tool to find electric fields in
problems involving material boundaries.
There are in fact two methods to develop the desired
differential equation. One method is via the definition
of divergence, whereas the other is via the divergence
theorem. Both methods are presented below because
each provides a different bit of insight. Let’s explore
the first method:
Derivation via the definition of divergence. Let the
geometrical volume enclosed by S be V, which has
volume V (units of m3 ). Dividing both sides of
Equation 5.64 by V and taking the limit as V → 0:
H
D · ds
Qencl
= lim
(5.65)
lim S
V →0 V
V →0
V
The quantity on the right hand side is the volume
charge density ρv (units of C/m3 ) at the point at
which we converge after letting the volume go to
zero. The left hand side is, by definition, the
divergence of D, indicated in mathematical notation
as “∇ · D” (see Section 4.6). Thus, we have Gauss’
Law in differential form:
∇ · D = ρv
(5.66)
To interpret this equation, recall that divergence is
simply the flux (in this case, electric flux) per unit
volume.
Gauss’ Law in differential form (Equation 5.66)
says that the electric flux per unit volume originating from a point in space is equal to the volume charge density at that point.
Derivation via the divergence theorem.
Equation 5.66 may also be obtained from
Equation 5.64 using the Divergence Theorem
(Section 4.7), which in the present case may be
written:
Z
I
V
(∇ · D) dv =
S
D · ds
(5.67)
From Equation 5.64, we see that the right hand side of
the equation may be replaced with the enclosed
charge:
Z
(∇ · D) dv = Qencl
(5.68)
V
Furthermore, the enclosed charge can be expressed as
an integration of the volume charge density ρv over V:
Z
Z
ρv dv
(5.69)
(∇ · D) dv =
V
V
The above relationship must hold regardless of the
specific location or shape of V. The only way this is
possible is if the integrands are equal. Thus,
∇ · D = ρv , and we have obtained Equation 5.66.
Example 5.6. Determining the charge density at
a point, given the associated electric field.
The electric field intensity in free space is
E(r) = x̂Ax2 + ŷBz + ẑCx2 z
where A = 3 V/m3 , B = 2 V/m2 , and
C = 1 V/m4 . What is the charge density at
r = x̂2 − ŷ2 m?
Solution. First, we use D = ǫE to get D. Since
the problem is in free space, ǫ = ǫ0 . Thus we
have that the volume charge density is
ρv = ∇ · D
= ∇ · (ǫ0 E) = ǫ0 ∇ · E
∂
∂
∂
2
2
= ǫ0
(Bz) +
Ax +
Cx z
∂x
∂y
∂z
= ǫ0 2Ax + 0 + Cx2
Now calculating the charge density at the
specified location r:
ǫ0 2(3 V/m3 )(2 m) + 0 + (1 V/m4 )(2 m)2
= ǫ0 (16 V/m)
= 142 pC/m3
To obtain the electric field from the charge
distribution in the presence of boundary conditions
imposed by materials and structure, we must enforce
the relevant boundary conditions. These boundary
conditions are presented in Sections 5.17 and 5.18.
Frequently, a simpler approach requiring only the
boundary conditions on the electric potential (V (r))
is possible; this is presented in Section 5.15.
Furthermore, the reader should note the following.
Gauss’ Law does not always necessarily fully
5.8. FORCE, ENERGY, AND POTENTIAL DIFFERENCE
constrain possible solutions for the electric field. For
that, we might also need Kirchoff’s Voltage Law; see
Section 5.11.
Before moving on, it is worth noting that
Equation 5.66 can be solved in the special case in
which there are no boundary conditions to satisfy;
i.e., for charge only, in a uniform and unbounded
medium. In fact, no additional electromagnetic theory
is required to do this. Here’s the solution:
Z
1
r − r′
ρv (r′ ) dv
(5.70)
D(r) =
4π V |r − r′ |3
which we recognize as one of the results obtained in
Section 5.4 (after dividing both sides by ǫ to get E). It
is reasonable to conclude that Gauss’ Law (in either
integral or differential form) is fundamental, whereas
Coulomb’s Law is merely a consequence of Gauss’
Law.
Additional Reading:
• “Gauss’ Law” on Wikipedia.
• “Partial differential equation” on Wikipedia.
• “Boundary value problem” on Wikipedia.
105
5.8 Force, Energy, and Potential
Difference
[m0061]
The force Fe experienced by a particle at location r
bearing charge q in an electric field intensity E is (see
Sections 2.2 and/or Section 5.1)
Fe = qE(r)
(5.71)
If left alone in free space, this particle would
immediately begin to move. The resulting
displacement represents a loss of potential energy.
This loss can quantified using the concept of work,
W . The incremental work ∆W done by moving the
particle a short distance ∆l, over which we assume
the change in Fe is negligible, is
∆W ≈ −Fe · l̂∆l
(5.72)
where in this case l̂ is the unit vector in the direction
of the motion; i.e., the direction of Fe . The minus
sign indicates that potential energy of the system
consisting of the electric field and the particle is being
reduced. Like a spring that was previously
compressed and is now released, the system is
“relaxing.”
To confirm that work defined in this way is an
expression of energy, consider the units. The product
of force (units of N) and distance (units of m) has
units of N·m, and 1 N·m is 1 J of energy.
Now, what if the motion of the particle is due to
factors other than the force associated with the
electric field? For example, we might consider
“resetting” the system to it’s original condition by
applying an external force to overcome Fe .
Equation 5.72 still represents the change in potential
energy of the system, but now l̂ changes sign. The
same magnitude of work is done, but now this work is
positive. In other words, positive work requires the
application of an external force that opposes and
overcomes the force associated with the electric field,
thereby increasing the potential energy of the system.
With respect to the analogy of a mechanical spring
used above, positive work is achieved by compressing
the spring.
CHAPTER 5. ELECTROSTATICS
106
It is also worth noting that the purpose of the dot
product in Equation 5.72 is to ensure that only the
component of motion parallel to the direction of the
electric field is included in the energy tally. This is
simply because motion in any other direction cannot
be due to E, and therefore does not increase or
decrease the associated potential energy.
point (1) and end point (2) of C. V21 is defined as the
work done by traversing C, per unit of charge:
V21 ,
(5.73)
Equation 5.73 gives the work only for a short distance
around r. Now let us try to generalize this result. If
we wish to know the work done over a larger
distance, then we must account for the possibility that
E varies along the path taken. To do this, we may
sum contributions from points along the path traced
out by the particle, i.e.,
W ≈
N
X
∆W (rn )
N
X
n=1
V21 = −
Z
C
E · dl
(5.79)
An advantage of analysis in terms of electrical
potential as opposed to energy is that we will no
longer have to explicitly state the value of the charge
involved.
The potential difference V21 between two points
in space, given by Equation 5.79, is the change
in potential energy of a charged particle divided
by the charge of the particle. Potential energy is
also commonly known as “voltage” and has units
of V.
(5.74)
n=1
where rn are positions defining the path. Substituting
Equation 5.73, we have
W ≈ −q
(5.78)
This has units of J/C, which is volts (V). Substituting
Equation 5.77, we obtain:
We can make the relationship between work and the
electric field explicit by substituting Equation 5.71
into Equation 5.72, yielding
∆W ≈ −qE(r) · l̂∆l
W
q
E(rn ) · l̂(rn )∆l
Taking the limit as ∆l → 0 we obtain
Z
W = −q E(r) · l̂(r)dl
(5.75)
(5.76)
C
where C is the path (previously, the sequence of rn ’s)
followed. Now omitting the explicit dependence on r
in the integrand for clarity:
Z
W = −q E · dl
(5.77)
C
where dl = l̂dl as usual. Now, we are able to
determine the change in potential energy for a
charged particle moving along any path in space,
given the electric field.
At this point, it is convenient to formally define the
electric potential difference V21 between the start
Example 5.7. Potential difference in a uniform
electric field.
Consider an electric field E(r) = ẑE0 , which is
constant in both magnitude and direction
throughout the domain of the problem. The path
of interest is a line beginning at ẑz1 and ending
at ẑz2 . What is V21 ? (It’s worth noting that the
answer to this problem is a building block for a
vast number of problems in electromagnetic
analysis.)
Solution. From Equation 5.79 we have
Z z2
(ẑE0 ) · ẑdz = −E0 (z2 − z1 )
V21 = −
z1
(5.80)
Note V21 is simply the electric field intensity
times the distance between the points. This may
seem familiar. For example, compare this to the
findings of the battery-charged capacitor
experiment described in Section 2.2. There too
we find that potential difference equals electric
field intensity times distance, and the signs
agree.
5.9. INDEPENDENCE OF PATH
The solution to the preceding example is simple
because the direct path between the two points is
parallel to the electric field. If the path between the
points had been perpendicular to E, then the solution
is even easier – V21 is simply zero. In all other cases,
V21 is proportional to the component of the direct
path between the start and end points that is parallel
to E, as determined by the dot product.
107
5.9 Independence of Path
[m0062]
In Section 5.8, we found that the potential difference
(“voltage”) associated with a path C in an electric
field intensity E is
Z
V21 = − E · dl
(5.81)
C
where the curve begins at point 1 and ends at point 2.
Let these points be identified using the position
vectors r1 and r2 , respectively.1 Then:
Z r2
E · dl
(5.82)
V21 = −
r1 , along C
The associated work done by a particle bearing
charge q is
W21 = qV21
(5.83)
This work represents the change in potential energy
of the system consisting of the electric field and the
charged particle. So, it must also be true that
W21 = W2 − W1
(5.84)
where W2 and W1 are the potential energies when the
particle is at r2 and r1 , respectively. It is clear from
the above equation that W21 does not depend on C; it
depends only on the positions of the start and end
points and not on any of the intermediate points along
C. That is,
V21 = −
Z r2
r1
E · dl , independent of C
(5.85)
Since the result of the integration in Equation 5.85 is
independent of the path of integration, any path that
begins at r1 and ends at r2 yields the same value of
W21 and V21 . We refer to this concept as
independence of path.
The integral of the electric field over a path between two points depends only on the locations
of the start and end points and is independent of
the path taken between those points.
1 See Section 4.1 for a refresher on this concept.
CHAPTER 5. ELECTROSTATICS
108
A practical application of this concept is that some
paths may be easier to use than others, so there may
be an advantage in computing the integral in
Equation 5.85 using some path other than the path
actually traversed.
5.10 Kirchoff’s Voltage Law for
Electrostatics: Integral
Form
[m0016]
As explained in Section 5.9, the electrical potential at
point r2 relative to r1 in an electric field E (V/m) is
Z r2
E · dl
(5.86)
V21 = −
r1
where the path of integration may be any path that
begins and ends at the specified points. Consider what
happens if the selected path through space begins and
ends at the same point; i.e., r2 = r1 . In this case, the
path of integration is a closed loop. Since V21
depends only on the positions of the start and end
points and because the potential energy at those
points is the same, we conclude:
I
E · dl = 0
(5.87)
This principle is known as Kirchoff’s Voltage Law for
Electrostatics.
Kirchoff’s Voltage Law for Electrostatics (Equation 5.87) states that the integral of the electric
field over a closed path is zero.
It is worth noting that this law is a generalization of a
principle of which the reader is likely already aware.
In electric circuit theory, the sum of voltages over any
closed loop in a circuit is zero. This is also known as
Kirchoff’s Voltage Law because it is precisely the
same principle. To obtain Equation 5.87 for an
electric circuit, simply partition the closed path into
branches, with each branch representing one
component. Then, the integral of E over each branch
is the branch voltage; i.e., units of V/m times units of
m yields units of V. Then, the sum of these branch
voltages over any closed loop is zero, as dictated by
Equation 5.87.
Finally, be advised that Equation 5.87 is specific to
electrostatics. In electrostatics, it is assumed that the
electric field is independent of the magnetic field.
This is true if the magnetic field is either zero or not
5.11. KIRCHOFF’S VOLTAGE LAW FOR ELECTROSTATICS: DIFFERENTIAL FORM
109
time-varying. If the magnetic field is time-varying,
then Equation 5.87 must be modified to account for
the effect of the magnetic field, which is to make the
right hand size potentially different from zero. The
generalized version of this expression that correctly
accounts for that effect is known as the
Maxwell-Faraday Equation (Section 8.8).
5.11 Kirchoff’s Voltage Law for
Electrostatics: Differential
Form
Additional Reading:
The integral form of Kirchoff’s Voltage Law for
electrostatics (KVL; Section 5.10) states that an
integral of the electric field along a closed path is
equal to zero:
I
• “Maxwell’s Equations” on Wikipedia.
• “Kirchoff’s Circuit Laws” on Wikipedia.
[m0152]
C
E · dl = 0
(5.88)
where E is electric field intensity and C is the closed
curve. In this section, we derive the differential form
of this equation. In some applications, this differential
equation, combined with boundary conditions
imposed by structure and materials (Sections 5.17 and
5.18), can be used to solve for the electric field in
arbitrarily complicated scenarios. A more immediate
reason for considering this differential equation is that
we gain a little more insight into the behavior of the
electric field, disclosed at the end of this section.
The equation we seek may be obtained using Stokes’
Theorem (Section 4.9), which in the present case may
be written:
Z
I
(∇ × E) · ds =
E · dl
(5.89)
S
C
where S is any surface bounded by C, and ds is the
normal to that surface with direction determined by
right-hand rule. The integral form of KVL tells us that
the right hand side of the above equation is zero, so:
Z
(∇ × E) · ds = 0
(5.90)
S
The above relationship must hold regardless of the
specific location or shape of S. The only way this is
possible for all possible surfaces is if the integrand is
zero at every point in space. Thus, we obtain the
desired expression:
∇×E=0
(5.91)
Summarizing:
The differential form of Kirchoff’s Voltage Law
for electrostatics (Equation 5.91) states that the
curl of the electrostatic field is zero.
CHAPTER 5. ELECTROSTATICS
110
Equation 5.91 is a partial differential equation. As
noted above, this equation, combined with the
appropriate boundary conditions, can be solved for
the electric field in arbitrarily-complicated scenarios.
Interestingly, it is not the only such equation available
for this purpose – Gauss’ Law (Section 5.7) also does
this. Thus, we see a system of partial differential
equations emerging, and one may correctly infer that
that the electric field is not necessarily fully
constrained by either equation alone.
5.12 Electric Potential Field Due
to Point Charges
Additional Reading:
In Sections 5.8 and 5.9, it was determined that the
potential difference measured from position r1 to
position r2 is
Z r2
E · dl
(5.93)
V21 = −
• “Maxwell’s Equations” on Wikipedia.
• “Boundary value problem” on Wikipedia.
[m0064]
The electric field intensity due to a point charge q at
the origin is (see Section 5.1 or 5.5)
E = r̂
q
4πǫr2
(5.92)
r1
This method for calculating potential difference is
often a bit awkward. To see why, consider an example
from circuit theory, shown in Figure 5.5. In this
example, consisting of a single resistor and a ground
node, we’ve identified four quantities:
• The resistance R
• The current I through the resistor
• The node voltage V1 , which is the potential
difference measured from ground to the left side
of the resistor
• The node voltage V2 , which is the potential
difference measured from ground to the right
side of the resistor
Let’s say we wish to calculate the potential difference
V21 across the resistor. There are two ways this can be
done:
• V21 = −IR
• V21 = V2 − V1
The advantage of the second method is that it is not
necessary to know I, R, or indeed anything about
what is happening between the nodes; it is only
necessary to know the node voltages. The point is that
it is often convenient to have a common datum – in
5.12. ELECTRIC POTENTIAL FIELD DUE TO POINT CHARGES
V1
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R
c K. Kikkeri CC BY SA 4.0
Figure 5.5: A resistor in a larger circuit, used as an
example to demonstrate the concept of node voltages.
this example, ground – with respect to which the
potential differences at all other locations of interest
can be defined. When we have this, calculating
potential differences reduced to simply subtracting
predetermined node potentials.
So, can we establish a datum in general electrostatic
problems that works the same way? The answer is
yes. The datum is arbitrarily chosen to be a sphere
that encompasses the universe; i.e., a sphere with
radius → ∞. Employing this choice of datum, we can
use Equation 5.93 to define V (r), the potential at
point r, as follows:
V (r) , −
Z r
∞
E · dl
(5.94)
The electrical potential at a point, given by Equation 5.94, is defined as the potential difference
measured beginning at a sphere of infinite radius
and ending at the point r. The potential obtained
in this manner is with respect to the potential infinitely far away.
In the particular case where E is due to the point
charge at the origin:
Z rh
q i
r̂
V (r) = −
· dl
(5.95)
4πǫr2
∞
The principle of independence of path (Section 5.9)
asserts that the path of integration doesn’t matter as
long as the path begins at the datum at infinity and
ends at r. So, we should choose the easiest such path.
The radial symmetry of the problem indicates that the
111
easiest path will be a line of constant θ and φ, so we
choose dl = r̂dr. Continuing:
Z rh
q i
· [r̂dr]
(5.96)
r̂
V (r) = −
4πǫr2
∞
Z r
1
q
dr
(5.97)
=−
4πǫ ∞ r2
r
q 1
=+
(5.98)
4πǫ r ∞
so
V (r) = +
q
4πǫr
(5.99)
(Suggestion: Confirm that Equation 5.99 is
dimensionally correct.) In the context of the circuit
theory example above, this is the “node voltage” at r
when the datum is defined to be the surface of a
sphere at infinity. Subsequently, we may calculate the
potential difference from any point r1 to any other
point r2 as
V21 = V (r2 ) − V (r1 )
(5.100)
and that will typically be a lot easier than using
Equation 5.93.
It is not often that one deals with systems consisting
of a single charged particle. So, for the above
technique to be truly useful, we need a
straightforward way to determine the potential field
V (r) for arbitrary distributions of charge. The first
step in developing a more general expression is to
determine the result for a particle located at a point r′
somewhere other than the origin. Since Equation 5.99
depends only on charge and the distance between the
field point r and r′ , we have
V (r; r′ ) , +
q′
4πǫ |r − r′ |
(5.101)
where, for notational consistency, we use the symbol
q ′ to indicate the charge. Now applying superposition,
the potential field due to N charges is
V (r) =
N
X
V (r; rn )
(5.102)
n=1
Substituting Equation 5.101 we obtain:
N
1 X qn
V (r) =
4πǫ n=1 |r − rn |
(5.103)
CHAPTER 5. ELECTROSTATICS
112
Equation 5.103 gives the electric potential at
a specified location due to a finite number of
charged particles.
The potential field due to continuous distributions of
charge is addressed in Section 5.13.
5.13 Electric Potential Field due
to a Continuous Distribution
of Charge
[m0065]
The electrostatic potential field at r associated with N
charged particles is
N
V (r) =
1 X qn
4πǫ n=1 |r − rn |
(5.104)
where qn and rn are the charge and position of the
nth particle. However, it is more common to have a
continuous distribution of charge as opposed to a
countable number of charged particles. We now
consider how to compute V (r) three types of these
commonly-encountered distributions. Before
beginning, it’s worth noting that the methods will be
essentially the same, from a mathematical viewpoint,
as those developed in Section 5.4; therefore, a review
of that section may be helpful before attempting this
section.
Continuous Distribution of Charge Along a Curve.
Consider a continuous distribution of charge along a
curve C. The curve can be divided into short segments
of length ∆l. Then, the charge associated with the
nth segment, located at rn , is
qn = ρl (rn ) ∆l
(5.105)
where ρl is the line charge density (units of C/m) at
rn . Substituting this expression into Equation 5.104,
we obtain
N
V(r) =
1 X ρl (rn )
∆l
4πǫ n=1 |r − rn |
Taking the limit as ∆l → 0 yields:
Z
1
ρl (l)
V (r) =
dl
4πǫ C |r − r′ |
(5.106)
(5.107)
where r′ represents the varying position along C with
integration along the length l.
Continuous Distribution of Charge Over a
Surface. Consider a continuous distribution of charge
5.14. ELECTRIC FIELD AS THE GRADIENT OF POTENTIAL
5.14 Electric Field as the
Gradient of Potential
over a surface S. The surface can be divided into
small patches having area ∆s. Then, the charge
associated with the nth patch, located at rn , is
qn = ρs (rn ) ∆s
(5.108)
where ρs is surface charge density (units of C/m2 ) at
rn . Substituting this expression into Equation 5.104,
we obtain
N
V (r) =
1 X ρs (rn )
∆s
4πǫ n=1 |r − rn |
Taking the limit as ∆s → 0 yields:
Z
1
ρs (r′ )
V (r) =
ds
4πǫ S |r − r′ |
(5.109)
(5.110)
where r′ represents the varying position over S with
integration.
Continuous Distribution of Charge in a Volume.
Consider a continuous distribution of charge within a
volume V. The volume can be divided into small cells
(volume elements) having area ∆v. Then, the charge
associated with the nth cell, located at rn , is
qn = ρv (rn ) ∆v
[m0063]
In Section 5.8, it was determined that the electrical
potential difference V21 measured over a path C is
given by
Z
V21 = −
where ρv is the volume charge density (units of C/m )
at rn . Substituting this expression into
Equation 5.104, we obtain
Taking the limit as ∆v → 0 yields:
Z
ρv (r′ )
1
dv
V (r) =
4πǫ V |r − r′ |
E(r) · dl
(5.114)
We begin by identifying the contribution of an
infinitesimal length of the integral to the total integral
in Equation 5.114. At point r, this is
dV = −E(r) · dl
(5.115)
Although we can proceed using any coordinate
system, the following derivation is particularly simple
in Cartesian coordinates. In Cartesian coordinates,
N
1 X ρv (rn )
∆v
4πǫ n=1 |r − rn |
C
where E(r) is the electric field intensity at each point
r along C. In Section 5.12, we defined the scalar
electric potential field V (r) as the electric potential
difference at r relative to a datum at infinity. In this
section, we address the “inverse problem” – namely,
how to calculate E(r) given V (r). Specifically, we
are interested in a direct “point-wise” mathematical
transform from one to the other. Since Equation 5.114
is in the form of an integral, it should not come as a
surprise that the desired expression will be in the form
of a differential equation.
(5.111)
3
V (r) =
113
dl = x̂dx + ŷdy + ẑdz
(5.112)
(5.116)
We also note that for any scalar function of position,
including V (r), it is true that
(5.113)
where r′ represents the varying position over V with
integration.
dV =
∂V
∂V
∂V
dx +
dy +
dz
∂x
∂y
∂z
(5.117)
Note the above relationship is not specific to
electromagnetics; it is simply mathematics. Also note
that dx = dl · x̂ and so on for dy and dz. Making
these substitutions into the above equation, we obtain:
∂V
∂V
∂V
(dl · x̂) +
(dl · ŷ) +
(dl · ẑ)
∂x
∂y
∂z
(5.118)
This equation may be rearranged as follows:
∂
∂
∂
dV =
V · dl (5.119)
x̂
+ ŷ
+ ẑ
∂x
∂y
∂z
dV =
CHAPTER 5. ELECTROSTATICS
114
Comparing the above equation to Equation 5.115, we
find:
∂
∂
∂
V
(5.120)
+ ŷ
+ ẑ
E(r) = − x̂
∂x
∂y
∂z
Note that the quantity in square brackets is the
gradient operator “∇” (Section 4.5). Thus, we may
write
E = −∇V
(5.121)
which is the relationship we seek.
The electric field intensity at a point is the gradient of the electric potential at that point after a
change of sign (Equation 5.121).
Using Equation 5.121, we can immediately find the
electric field at any point r if we can describe V as a
function of r. Furthermore, this relationship between
V and E has a useful physical interpretation. Recall
that the gradient of a scalar field is a vector that points
in the direction in which that field increases most
quickly. Therefore:
The electric field points in the direction in which
the electric potential most rapidly decreases.
we found the scalar potential for this source was:
V (r) =
Example 5.8. Electric field of a charged
particle, beginning with the potential field.
In this example, we determine the electric field
of a particle bearing charge q located at the
origin. This may be done in a “direct” fashion
using Coulomb’s Law (Section 5.1). However,
here we have the opportunity to find the electric
field using a different method. In Section 5.12
(5.122)
So, we may obtain the electric field using
Equation 5.121:
q (5.123)
E = −∇V = −∇
4πǫr
Here V (r) is expressed in spherical coordinates,
so we have (Section B.2):
∂
∂ q 1 ∂
1
E = − r̂
+ θ̂
+ φ̂
∂r
r ∂θ
r sin θ ∂φ
4πǫr
(5.124)
In this case, V (r) does not vary with φ or θ, so
the second and third terms of the gradient are
zero. This leaves
∂ q E = −r̂
∂r 4πǫr
q ∂ 1
= −r̂
4πǫ ∂r
r q
1
= −r̂
− 2
(5.125)
4πǫ
r
So we find
E = +r̂
This result should not come as a complete surprise;
for example, the reader should already be aware that
the electric field points away from regions of net
positive charge and toward regions of net negative
charge (Sections 2.2 and/or 5.1). What is new here is
that both the magnitude and direction of the electric
field may be determined given only the potential field,
without having to consider the charge that is the
physical source of the electrostatic field.
q
4πǫr
q
4πǫr2
as was determined in Section 5.1.
(5.126)
5.15. POISSON’S AND LAPLACE’S EQUATIONS
5.15 Poisson’s and Laplace’s
Equations
[m0067]
The electric scalar potential field V (r), defined in
Section 5.12, is useful for a number of reasons
including the ability to conveniently compute
potential differences (i.e., V21 = V (r2 ) − V (r1 )) and
the ability to conveniently determine the electric field
by taking the gradient (i.e., E = −∇V ). One way to
obtain V (r) is by integration over the source charge
distribution, as described in Section 5.13. This
method is awkward in the presence of material
interfaces, which impose boundary conditions on the
solutions that must be satisfied simultaneously. For
example, the electric potential on a perfectly
conducting surface is constant2 – a constraint which is
not taken into account in any of the expressions in
Section 5.13.
In this section, we develop an alternative approach to
calculating V (r) that accommodates these boundary
conditions, and thereby facilitates the analysis of the
scalar potential field in the vicinity of structures and
spatially-varying material properties. This alternative
approach is based on Poisson’s Equation, which we
now derive.
We begin with the differential form of Gauss’ Law
(Section 5.7):
∇ · D = ρv
(5.127)
Using the relationship D = ǫE (and keeping in mind
our standard assumptions about material properties,
summarized in Section 2.8) we obtain
ρv
(5.128)
∇·E=
ǫ
Next, we apply the relationship (Section 5.14):
E = −∇V
(5.129)
yielding
ρv
(5.130)
∇ · ∇V = −
ǫ
This is Poisson’s Equation, but it is not in the form in
which it is commonly employed. To obtain the
2 This fact is probably already known to the reader from past
study of elementary circuit theory; however, this is established in
the context of electromagnetics in Section 5.19.
115
alternative form, consider the operator ∇ · ∇ in
Cartesian coordinates:
∂
∂
∂
∂
∂
∂
x̂ +
ŷ +
ẑ ·
x̂ +
ŷ +
ẑ
∇·∇=
∂x
∂y
∂z
∂x
∂y
∂z
2
2
2
∂
∂
∂
+ 2+ 2
=
∂x2
∂y
∂z
= ∇2
(5.131)
i.e., the operator ∇ · ∇ is identically the Laplacian
operator ∇2 (Section 4.10). Furthermore, this is true
regardless of the coordinate system employed. Thus,
we obtain the following form of Poisson’s Equation:
∇2 V = −
ρv
ǫ
(5.132)
Poisson’s Equation (Equation 5.132) states that
the Laplacian of the electric potential field is
equal to the volume charge density divided by the
permittivity, with a change of sign.
Note that Poisson’s Equation is a partial differential
equation, and therefore can be solved using
well-known techniques already established for such
equations. In fact, Poisson’s Equation is an
inhomogeneous differential equation, with the
inhomogeneous part −ρv /ǫ representing the source of
the field. In the presence of material structure, we
identify the relevant boundary conditions at the
interfaces between materials, and the task of finding
V (r) is reduced to the purely mathematical task of
solving the associated boundary value problem (see
“Additional Reading” at the end of this section). This
approach is particularly effective when one of the
materials is a perfect conductor or can be modeled as
such a material. This is because – as noted at the
beginning of this section – the electric potential at all
points on the surface of a perfect conductor must be
equal, resulting in a particularly simple boundary
condition.
In many other applications, the charge responsible for
the electric field lies outside the domain of the
problem; i.e., we have non-zero electric field (hence,
potentially non-zero electric potential) in a region that
is free of charge. In this case, Poisson’s Equation
simplifies to Laplace’s Equation:
∇2 V = 0 (source-free region)
(5.133)
CHAPTER 5. ELECTROSTATICS
116
Laplace’s Equation (Equation 5.133) states that
the Laplacian of the electric potential field is zero
in a source-free region.
5.16 Potential Field Within a
Parallel Plate Capacitor
[m0068]
Like Poisson’s Equation, Laplace’s Equation,
combined with the relevant boundary conditions, can
be used to solve for V (r), but only in regions that
contain no charge.
Additional Reading:
• “Poisson’s equation” on Wikipedia.
• “Boundary value problem” on Wikipedia.
• “Laplace’s equation” on Wikipedia.
This section presents a simple example that
demonstrates the use of Laplace’s Equation
(Section 5.15) to determine the potential field in a
source free region. The example, shown in Figure 5.6,
pertains to an important structure in electromagnetic
theory – the parallel plate capacitor. Here we are
concerned only with the potential field V (r) between
the plates of the capacitor; you do not need to be
familiar with capacitance or capacitors to follow this
section (although you’re welcome to look ahead to
Section 5.22 for a preview, if desired). What is
recommended before beginning is a review of the
battery-charged capacitor experiment discussed in
Section 2.2. In this section you’ll see a rigorous
derivation of what we figured out in an informal way
in that section.
The parallel-plate capacitor in Figure 5.6 consists of
two perfectly-conducting circular disks separated by a
distance d by a spacer material having permittivity ǫ.
There is no charge present in the spacer material, so
Laplace’s Equation applies. That equation is
(Section 5.15):
∇2 V = 0 (source-free region)
(5.134)
Let VC be the potential difference between the plates,
which would also be the potential difference across
the terminals of the capacitor. The radius a of the
plates is larger than d by enough that we may neglect
what is going on at at the edges of the plates – more
on this will be said as we work the problem. Under
this assumption, what is the electric potential field
V (r) between the plates?
This problem has cylindrical symmetry, so it makes
sense to continue to use cylindrical coordinates with
the z axis being perpendicular to the plates.
Equation 5.134 in cylindrical coordinates is:
1 ∂
∂
1 ∂2
∂2
ρ
+ 2 2 + 2 V = 0 (5.135)
ρ ∂ρ
∂ρ
ρ ∂φ
∂z
or perhaps a little more clearly written as follows:
1 ∂
∂V
1 ∂2V
∂2V
ρ
+ 2
+
=0
(5.136)
ρ ∂ρ
∂ρ
ρ ∂φ2
∂z 2
5.16. POTENTIAL FIELD WITHIN A PARALLEL PLATE CAPACITOR
z
above yields the expected result for the electric field
intensity: E ≈ −ẑVC /d. This is precisely the result
that we arrived at (without the aid of Laplace’s
Equation) in Section 2.2.
V-+VC
d
ρ
a
V-
Figure 5.6: A parallel plate capacitor, as a demonstration of the use of Laplace’s Equation.
Since the problem has radial symmetry, ∂V /∂φ = 0.
Since d ≪ a, we expect the fields to be approximately
constant with ρ until we get close to the edge of the
plates. Therefore, we assume ∂V /∂ρ is negligible and
can be taken to be zero. Thus, we are left with
∂2V
≈ 0 for ρ ≪ a
∂z 2
(5.137)
The general solution to Equation 5.137 is obtained
simply by integrating both sides twice, yielding
V (z) = c1 z + c2
(5.138)
where c1 and c2 are constants that must be consistent
with the boundary conditions. Thus, we must develop
appropriate boundary conditions. Let the node
voltage at the negative (z = 0) terminal be V− . Then
the voltage at the positive (z = +d) terminal is
V− + VC . Therefore:
V (z = 0) = V−
V (z = +d) = V− + VC
(5.139)
(5.140)
These are the relevant boundary conditions.
Substituting V (z = 0) = V− into Equation 5.138
yields c2 = V− . Substituting V (z = +d) = V− + VC
into Equation 5.138 yields c1 = VC /d. Thus, the
answer to the problem is
V (z) ≈
VC
z + V− for ρ ≪ a
d
117
(5.141)
Note that the above result is dimensionally correct
and confirms that the potential deep inside a “thin”
parallel plate capacitor changes linearly with distance
between the plates.
Further, you should find that application of the
equation E = −∇V (Section 5.14) to the solution
A reasonable question to ask at this point would be,
what about the potential field close to the edge of the
plates, or, for that matter, beyond the plates? The field
in this region is referred to as a fringing field. For the
fringing field, ∂V /∂ρ is no longer negligible and must
be taken into account. In addition, it is necessary to
modify the boundary conditions to account for the
outside surfaces of the plates (that is, the sides of the
plates that face away from the dielectric) and to
account for the effect of the boundary between the
spacer material and free space. These issues make the
problem much more difficult. When an accurate
calculation of a fringing field is necessary, it is
common to resort to a numerical solution of Laplace’s
Equation. Fortunately, accurate calculation of
fringing fields is usually not required in practical
engineering applications.
CHAPTER 5. ELECTROSTATICS
118
5.17 Boundary Conditions on the
Electric Field Intensity (E)
A
t
B
w
[m0020]
In homogeneous media, electromagnetic quantities
vary smoothly and continuously. At an interface
between dissimilar media, however, it is possible for
electromagnetic quantities to be discontinuous. These
discontinuities can be described mathematically as
boundary conditions and used to to constrain
solutions for the associated electromagnetic
quantities. In this section, we derive boundary
conditions on the electric field intensity E.
To begin, consider a region consisting of only two
media that meet at an interface defined by the
mathematical surface S, as shown in Figure 5.7. If
either one of the materials is a perfect electrical
conductor (PEC), then S is an equipotential surface;
i.e., the electric potential V is constant everywhere on
S. Since E is proportional to the spatial rate of
change of potential (recall E = −∇V ; Section 5.14),
we find:
The component of E that is tangent to a perfectlyconducting surface is zero.
This is sometimes expressed informally as follows:
Etan = 0 on PEC surface
(5.142)
where “Etan ” is understood to be the component of E
that is tangent to S. Since the tangential component
E
E
E
E
PEC
(equipotential)
c K. Kikkeri CC BY SA 4.0
Figure 5.7: At the surface of a perfectly-conducting
region, E may be perpendicular to the surface (two
leftmost possibilities), but may not exhibit a component that is tangent to the surface (two rightmost possibilities).
c K. Kikkeri CC BY SA 4.0
Figure 5.8: Use of KVL to determine the boundary
condition on E.
of E on the surface of a perfect conductor is zero, the
electric field at the surface must be oriented entirely
in the direction perpendicular to the surface, as shown
in Figure 5.7.
The following equation expresses precisely the same
idea, but includes the calculation of the tangential
component as part of the statement:
E × n̂ = 0 (on PEC surface)
(5.143)
where n̂ is either normal (i.e., unit vector
perpendicular to the surface) to each point on S. This
expression works because the cross product of any
two vectors is perpendicular to either vector
(Section 4.1), and any vector which is perpendicular
to n̂ is tangent to S.
We now determine a more general boundary
condition that applies even when neither of the media
bordering S is a perfect conductor. The desired
boundary condition can be obtained directly from
Kirchoff’s Voltage Law (KVL; Section 5.10):
I
C
E · dl = 0
(5.144)
Let the closed path of integration take the form of a
rectangle centered on S, as shown in Figure 5.8. Let
the sides A, B, C, and D be perpendicular or parallel
to the surface, respectively. Let the length of the
perpendicular sides be w, and let the length of the
5.17. BOUNDARY CONDITIONS ON THE ELECTRIC FIELD INTENSITY (E)
parallel sides be l. From KVL we have
I
Z
E · dl =
E · dl
C
ZA
+
E · dl
ZB
+
E · dl
ZC
+
E · dl
=0
or, as it is more commonly written:
n̂ × (E1 − E2 ) = 0 on S
(5.145)
Now, let us reduce w and l together while (1)
maintaining a constant ratio w/l ≪ 1 and (2) keeping
C centered on S. In this process, the contributions
from the B and D segments become equal in
magnitude but opposite in sign; i.e.,
Z
Z
E · dl +
E · dl → 0
(5.146)
D
This leaves
I
Z
Z
E · dl →
E · dl +
E · dl → 0
C
A
(5.147)
C
Let us define the unit vector t̂ (“tangent”) as shown in
Figure 5.8. When the lengths of sides A and C
become sufficiently small, we can write the above
expression as follows:
E1 · t̂∆l − E2 · t̂∆l → 0
(5.148)
where E1 and E2 are the fields evaluated on the two
sides of the boundary and ∆l → 0 is the length of
sides A and C while this is happening. Note that the
only way Equation 5.148 can be true is if the
tangential components of E1 and E2 are equal. In
other words:
The tangential component of E must be continuous across an interface between dissimilar media.
Note that this is a generalization of the result we
obtained earlier for the case in which one of the
media was a PEC – in that case, the tangent
component of E on the other side of the interface
must be zero because it is zero in the PEC medium.
As before, we can express this idea in compact
mathematical notation. Using the same idea used to
obtain Equation 5.143, we have found
E1 × n̂ = E2 × n̂ on S
(5.150)
We conclude this section with a note about the
broader applicability of this boundary condition:
D
B
119
(5.149)
Equation 5.150 is the boundary condition that applies to E for both the electrostatic and the general (time-varying) case.
Although a complete explanation is not possible
without the use of the Maxwell-Faraday Equation
(Section 8.8), the reason why this boundary condition
applies in the time-varying case can be disclosed here.
In the presence of time-varying magnetic fields, the
right-hand side of Equation 5.144 may become
non-zero and is proportional to the area defined by the
closed loop. However, the above derivation requires
the area of this loop to approach zero, in which case
the possible difference from Equation 5.144 also
converges to zero. Therefore, the boundary condition
expressed in Equation 5.150 applies generally.
CHAPTER 5. ELECTROSTATICS
120
5.18 Boundary Conditions on the
Electric Flux Density (D)
a
region 1
n
h
[m0021]
h
In this section, we derive boundary conditions on the
electric flux density D. The considerations are quite
similar to those encountered in the development of
boundary conditions on the electric field intensity (E)
in Section 5.17, so the reader may find it useful to
review that section before attempting this section.
This section also assumes familiarity with the
concepts of electric flux, electric flux density, and
Gauss’ Law; for a refresher, Sections 2.4 and 5.5 are
suggested.
To begin, consider a region at which two
otherwise-homogeneous media meet at an interface
defined by the mathematical surface S, as shown in
Figure 5.9. Let one of these regions be a perfect
electrical conductor (PEC). In Section 5.17, we
established that the tangential component of the
electric field must be zero, and therefore, the electric
field is directed entirely in the direction perpendicular
to the surface. We further know that the electric field
within the conductor is identically zero. Therefore, D
at any point on S is entirely in the direction
perpendicular to the surface and pointing into the
non-conducting medium. However, it is also possible
to determine the magnitude of D. We shall
demonstrate in this section that
At the surface of a perfect conductor, the magnitude of D is equal to the surface charge density
ρs (units of C/m2 ) at that point.
D=nρs
PEC
c K. Kikkeri CC BY SA 4.0
Figure 5.9: The component of D that is perpendicular
to a perfectly-conducting surface is equal to the charge
density on the surface.
region 2
c K. Kikkeri CC BY SA 4.0
Figure 5.10: Use of Gauss’ Law to determine the
boundary condition on D.
The following equation expresses precisely the same
idea, but includes the calculation of the perpendicular
component as part of the statement:
D · n̂ = ρs (on PEC surface)
(5.151)
where n̂ is the normal to S pointing into the
non-conducting region. (Note that the orientation of n̂
is now important; we have assumed n̂ points into
region 1, and we must now stick with this choice.)
Before proceeding with the derivation, it may be
useful to note that this result is not surprising. The
very definition of electric flux (Section 2.4) indicates
that D should correspond in the same way to a
surface charge density. However, we can show this
rigorously, and in the process we can generalize this
result to the more-general case in which neither of the
two materials are PEC.
The desired more-general boundary condition may be
obtained from the integral form of Gauss’ Law
(Section 5.5), as illustrated in Figure 5.10. Let the
surface of integration S ′ take the form of closed
cylinder centered at a point on the interface and for
which the flat ends are parallel to the surface and
perpendicular to n̂. Let the radius of this cylinder be
a, and let the length of the cylinder be 2h. From
Gauss’ Law we have
I
Z
D · ds =
D · ds
S′
top
Z
+
D · ds
Zside
D · ds = Qencl (5.152)
+
bottom
5.18. BOUNDARY CONDITIONS ON THE ELECTRIC FLUX DENSITY (D)
where the “top” and “bottom” are in Regions 1 and 2,
respectively, and Qencl is the charge enclosed by S ′ .
Now let us reduce h and a together while (1)
maintaining a constant ratio h/a ≪ 1 and (2) keeping
S ′ centered on S. Because h ≪ a, the area of the side
can be made negligible relative to the area of the top
and bottom. Then as h → 0 we are left with
Z
Z
D · ds +
D · ds → Qencl
(5.153)
top
bottom
As the area of the top and bottom sides become
infinitesimal, the variation in D over these areas
becomes negligible. Now we have simply:
D1 · n̂∆A + D2 · (−n̂) ∆A → Qencl
(5.154)
where D1 and D2 are the electric flux density vectors
in medium 1 and medium 2, respectively, and ∆A is
the area of the top and bottom sides. The above
expression can be rewritten
n̂ · (D1 − D2 ) →
Qencl
∆A
(5.155)
Note that the left side of the equation must represent a
actual, physical surface charge; this is apparent from
dimensional analysis and the fact that h is now
infinitesimally small. Therefore:
n̂ · (D1 − D2 ) = ρs
(5.156)
where, as noted above, n̂ points into region 1.
Summarizing:
Any discontinuity in the normal component of
the electric flux density across the boundary between two material regions is equal to the surface
charge.
Now let us verify that this is consistent with our
preliminary finding, in which Region 2 was a PEC. In
that case D2 = 0, so we see that Equation 5.151 is
satisfied, as expected. If neither Region 1 nor
Region 2 is PEC and there is no surface charge on the
interface, then we find n̂ · (D1 − D2 ) = 0; i.e.,
In the absence of surface charge, the normal component of the electric flux density must be continuous across the boundary.
121
Finally, we note that since D = ǫE, Equation 5.156
implies the following boundary condition on E:
n̂ · (ǫ1 E1 − ǫ2 E2 ) = ρs
(5.157)
where ǫ1 and ǫ2 are the permittivities in Regions 1
and 2, respectively. The above equation illustrates one
reason why we sometimes prefer the “flux”
interpretation of the electric field to the “field
intensity” interpretation of the electric field.
CHAPTER 5. ELECTROSTATICS
122
5.19 Charge and Electric Field
for a Perfectly Conducting
Region
[m0025]
In this section, we consider the behavior of charge
and the electric field in the vicinity of a perfect
electrical conductor (PEC).
First, note that the electric field – both the electric
field intensity E and electric flux density D –
throughout a PEC region is zero. This is because the
electrical potential throughout a PEC region must be
constant. (This idea is explored further in
Section 6.3.) Recall that the electric field is
proportional to the spatial rate of change of electrical
potential (i.e., E = −∇V ; Section 5.14). Thus, the
electric field must be zero throughout a PEC region.
Second, the electric field is oriented directly away
from (i.e., perpendicular to) the PEC surface, and the
magnitude of D is equal to the surface charge density
ρs (C/m2 ) (Section 5.18).
Now we address the question of charge distribution;
i.e., the location and density of charge. Consider the
scenario shown in Figure 5.11. Here, a flat slab of
PEC material is embedded in dielectric material.3 The
3 For the purposes of this section, it suffices to interpret “dielec-
thickness of the slab is finite, whereas the length and
width of the slab is infinite. The region above the slab
is defined as Region 1 and has permittivity ǫ1 . The
region below the slab is defined as Region 2 and has
permittivity ǫ2 . Electric fields E1 and E2 are present
in Regions 1 and 2, respectively, as shown in
Figure 5.11. To begin, let us assume that these fields
are the result of some external stimulus that results in
the direction of these fields being generally upward,
as shown in Figure 5.11.
Now, what do we know about E1 and E2 ? First, both
fields must satisfy the relevant boundary conditions.
That is, the component of E1 that is tangent to the
upper PEC surface is zero, so that E1 is directed
entirely in a direction perpendicular to the surface.
Similarly, the component of E2 that is tangent to the
lower PEC surface is zero, so that E2 is directed
entirely in a direction perpendicular to the surface. At
this point we have not determined the magnitudes or
signs of E1 and E2 ; we have established only that
there are no non-zero components tangential to (i.e.,
parallel to) the PEC surfaces.
Next, recall that the electric field must be zero within
the slab. This means that there must be zero net
charge within the slab, since any other distribution of
charge will result in a non-zero electric field, and
subsequently a potential difference between locations
within the slab. Therefore:
There can be no static charge within a PEC.
tric” as a “nonconducting and well characterized entirely in terms of
It follows that
Charge associated with a PEC lies entirely on the
surface.
c K. Kikkeri CC BY SA 4.0
Figure 5.11: An infinite flat slab of PEC in the presence of an applied electric field.
Outside the slab, the boundary conditions on D1 and
D2 in the dielectric regions require these fields to be
non-zero when the surface charge density on the PEC
is non-zero. The surface charge supports the
discontinuity in the normal component of the electric
fields. Specifically, D1 and D2 have the same
magnitude |ρs | because the surface charge densities
on both sides of the slab have equal magnitude.
However, the electric field intensity E1 = D1 /ǫ1 ,
whereas E2 = D2 /ǫ2 ; i.e., these are different. That
its permittivity.” For more, see Section 5.20.
5.20. DIELECTRIC MEDIA
123
5.20 Dielectric Media
+
+ +
+
++
+
+
+
+
[m0107]
+
−− − − −
−
−
−
−
−
+
+
+
−
+
− − − −−
−−
−
++ + +
++
by C. Burks (modified)
Figure 5.12: Electric field lines due to a point charge
in the vicinity of PEC regions (shaded) of various
shapes.
is, the electric field intensities are unequal unless the
permittivities in each dielectric region are equal.
Finally, let us consider the structure of the electric
field in more general cases. Figure 5.12 shows field
lines in a homogeneous dielectric material in which a
point charge and PEC regions of various shapes are
embedded. Note that electric field lines now bend in
the dielectric so as to satisfy the requirement that the
tangential component of the electric field be zero on
PEC surfaces. Also note that the charge distribution
arranges itself on the PEC surfaces so as to maintain
zero electric field and constant potential within the
cube.
Dielectric is particular category of materials that
exhibit low conductivity4 because their constituent
molecules remain intact when exposed to an electric
field, as opposed to shedding electrons as is the case
in good conductors. Subsequently, dielectrics do not
effectively pass current, and are therefore considered
“good insulators” as well as “poor conductors.” An
important application of dielectrics in electrical
engineering is as a spacer material in printed circuit
boards (PCBs), coaxial cables, and capacitors.
Examples of dielectrics include air, glass, teflon, and
fiberglass epoxy (the material used in common “FR4”
printed circuit boards). These and other dielectrics are
listed along with values of their constitutive
parameters in Section A.1.
The electromagnetic properties of dielectric materials
are quantified primarily by relative permittivity ǫr
(Section 2.3), which ranges from very close to 1
upward to roughly 50, and is less than 6 or so for
most commonly-encountered materials having low
moisture content. The permeability of dielectric
materials is approximately equal to the free-space
value (i.e., µ ≈ µ0 ); therefore, these materials are said
to be “non-magnetic.”
Additional Reading:
• “Dielectric” on Wikipedia.
4 See Section 2.8 for a refresher on parameters describing properties of materials.
CHAPTER 5. ELECTROSTATICS
124
5.21 Dielectric Breakdown
5.22 Capacitance
[m0109]
[m0112]
The permittivity of an ideal dielectric is independent
of the magnitude of an applied electric field; the
material is said to be “linear.”5 However, all practical
dielectrics fail in this respect with sufficiently strong
electric field. Typically, the failure is abrupt and is
observed as a sudden, dramatic increase in
conductivity, signaling that electrons are being
successfully dislodged from their host molecules. The
threshold value of the electric field intensity at which
this occurs is known as the dielectric strength, and the
sudden change in behavior observed in the presence
of an electric field greater than this threshold value is
known as dielectric breakdown.
When separate regions of positive and negative charge
exist in proximity, Coulomb forces (Section 5.1) will
attempt to decrease the separation between the
charges. As noted in Section 5.8, this can be
interpreted as a tendency of a system to reduce its
potential energy. If the charges are fixed in place, then
the potential energy remains constant. This potential
energy is proportional to the Coulomb force.
Referring back to Section 5.1, the Coulomb force is:
Dielectric strength varies from about 3 MV/m for air
to about 200 MV/m in mica (a dielectric commonly
used in capacitors).
• Inversely proportional to the separation between
the charges squared
Dielectric breakdown is typically accompanied by
“arcing,” which is a sudden flow of current associated
with the breakdown. A well known example of this
phenomenon is lightning, which occurs when charge
is exchanged between sky and ground when air (a
dielectric) exhibits breakdown. Dielectric breakdown
in solids typically damages the material.
Additional Reading:
• “Electrical Breakdown” on Wikipedia.
• Proportional to quantity of positive charge
squared
• Inversely proportional to the permittivity of the
material separating the charges
Therefore, the potential energy of the system is
likewise dependent on charge, separation, and
permittivity. Furthermore, we see that the ability of a
system to store energy in this manner depends on the
geometry of the charge distribution and the
permittivity of the intervening material.
Now recall that the electric field intensity E is
essentially defined in terms of the Coulomb force;
i.e., F = qE (Section 2.2). So, rather than thinking of
the potential energy of the system as being associated
with the Coulomb force, it is equally valid to think of
the potential energy as being stored in the electric
field associated with the charge distribution. It
follows from the previous paragraph that the energy
stored in the electric field depends on the geometry of
the charge distribution and the permittivity of the
intervening media. This relationship is what we mean
by capacitance. We summarize as follows:
Capacitance is the ability of a structure to store
energy in an electric field.
and
5 See Section 2.8 for a review of this concept.
5.22. CAPACITANCE
125
øùúûü
ýþÿcg
The capacitance of a structure depends on its geometry and the permittivity of the medium separating regions of positive and negative charge.
Note that capacitance does not depend on charge,
which we view as either a stimulus or response from
this point of view. The corresponding response or
stimulus, respectively, is the potential associated with
this charge. This leads to the following definition:
C,
Q+
V
(5.158)
where Q+ (units of C) is the total positive charge, V
(units of V) is the potential associated with this
charge (defined such that it is positive), and C (units
of F) is the associated capacitance. So:
In practice, capacitance is defined as the ratio
of charge present on one conductor of a twoconductor system to the potential difference between the conductors (Equation 5.158).
In other words, a structure is said to have greater
capacitance if it stores more charge – and therefore
stores more energy – in response to a given potential
difference.
Figure 5.13 shows the relevant features of this
definition. Here, a battery imposes the potential
difference V between two regions of
perfectly-conducting material. Q+ is the total charge
on the surface of the PEC region attached to the
positive terminal of the battery. An equal amount of
negative charge appears on the surface of the PEC
region attached to the negative terminal of the battery
(Section 5.19). This charge distribution gives rise to
an electric field. Assuming the two PEC regions are
fixed in place, Q+ will increase linearly with
increasing V , at a rate determined by the capacitance
C of the structure.
A capacitor is a device that is designed to exhibit a
specified capacitance. We can now make the
connection to the concept of the capacitor as it
appears in elementary circuit theory. In circuit theory,
the behavior of devices is characterized in terms of
terminal voltage VT in response to terminal current
IT , and vice versa. First, note that current does not
V
÷ñ ô
òó
ð
-
- -- PEC
ö õ
Q+
E
- t
- Q+
Figure 5.13:
tance.
Electrostatic interpretation of capaci-
normally flow through a capacitor,6 so when we refer
to “terminal current” for a capacitor, what we really
mean is the flow of charge arriving or departing from
one of the conductors via the circuit, which is equal to
the flow of charge departing or arriving (respectively)
at the other conductor. This gives the appearance of
current flow through the capacitor when the current is
examined from outside the capacitor. With that
settled, we proceed as follows. Using Equation 5.158,
we express the voltage VT across the terminals of a
capacitor having capacitance C:
VT =
Q+
C
(5.159)
We seek a relationship between VT and IT . Current is
charge per unit time, so the charge on either
conductor is the integral of IT over time; i.e.:
Q+ (t) =
Z t
IT (τ ) dτ + Q+ (t0 )
(5.160)
t0
where t0 is an arbitrarily-selected start time. In other
words, amps integrated over time is charge. If we
define IT as being positive in the direction of the flow
of positive charge as is the usual convention, then we
have:
Z
1 t
1
VT (t) =
IT (τ ) dτ + Q+ (t0 )
(5.161)
C t0
C
Again applying Equation 5.158, we see that the
second term is simply VT (t0 ). This is the expected
relationship from elementary circuit theory.
6 If it does, it’s probably experiencing dielectric breakdown; see
Section 5.21.
CHAPTER 5. ELECTROSTATICS
126
Finally, solving for IT we obtain the differential form
of this relationship:
IT (t) = C
d
VT (t)
dt
5.23 The Thin Parallel Plate
Capacitor
(5.162)
[m0070]
Additional Reading:
• “Capacitance” on Wikipedia.
• “Capacitor” on Wikipedia.
Let us now determine the capacitance of a common
type of capacitor known as the thin parallel plate
capacitor, shown in Figure 5.14. This capacitor
consists of two flat plates, each having area A,
separated by distance d. To facilitate discussion, let us
place the origin of the coordinate system at the center
of the lower plate, with the +z axis directed toward
the upper plate such that the upper plate lies in the
z = +d plane.
Below we shall find the capacitance by assuming a
particular charge on one plate, using the boundary
condition on the electric flux density D to relate this
charge density to the internal electric field, and then
integrating over the electric field between the plates to
obtain the potential difference. Then, capacitance is
the ratio of the assumed charge to the resulting
potential difference.
The principal difficulty in this approach is finding the
electric field. To appreciate the problem, first consider
that if the area of the plates was infinite, then the
electric field would be very simple; it would begin at
the positively-charged plate and extend in a
perpendicular direction toward the negatively-charged
plate (Section 5.19). Furthermore, the field would be
constant everywhere between the plates. This much is
apparent from symmetry alone. However, when the
plate area is finite, then we expect a fringing field to
emerge. “Fringing field” is simply a term applied to
the non-uniform field that appears near the edge of
the plates. The field is non-uniform in this region
z
d
ρ
Figure 5.14: Thin parallel plate capacitor.
5.23. THE THIN PARALLEL PLATE CAPACITOR
127
because the boundary conditions on the outside
(outward-facing) surfaces of the plates have a
significant effect in this region. In the central region
of the capacitor, however, the field is not much
different from the field that exists in the case of
infinite plate area.
surface is +ẑρs,− . Since +ẑρs,− = −ẑρs,+ , D
on the facing sides of the plates is equal.
In any parallel plate capacitor having finite plate area,
some fraction of the energy will be stored by the
approximately uniform field of the central region, and
the rest will be stored in the fringing field. We can
make the latter negligible relative to the former by
making the capacitor very “thin,” in the sense that the
smallest identifiable dimension of the plate is much
greater than d. Under this condition, we may obtain a
good approximation of the capacitance by simply
neglecting the fringing field, since an insignificant
fraction of the energy is stored there.
Imposing the “thin” condition leads to three
additional simplifications. First, the surface charge
distribution may be assumed to be approximately
uniform over the plate, which greatly simplifies the
analysis. Second, the shape of the plates becomes
irrelevant; they might be circular, square, triangular,
etc. When computing capacitance in the “thin” case,
only the plate area A is important. Third, the
thickness of each of the plates becomes irrelevant.
We are now ready to determine the capacitance of the
thin parallel plate capacitor. Here are the steps:
1. Assume a total positive charge Q+ on the upper
plate.
2. Invoking the “thin” condition, we assume the
charge density on the plates is uniform. Thus,
the surface charge density on bottom side of the
upper plate is ρs,+ = Q+ /A (C/m2 ).
3. From the boundary condition on the bottom
surface of the upper plate, D on this surface is
−ẑρs,+ .
4. The total charge on the lower plate, Q− , must be
equal and opposite the total charge on the upper
plate; i.e, Q− = −Q+ . Similarly, the surface
charge density on the upper surface of the lower
plate, ρs,− , must be −ρs,+ .
5. From the boundary condition on the top surface
of the lower plate (Section 5.18), D on this
6. Again invoking the “thin” condition, we assume
D between the plates has approximately the
same structure as we would see if the plate area
was infinite. Therefore, we are justified in
assuming D ≈ −ẑρs,+ everywhere between the
plates. (You might also see that this is
self-evident from the definition of D as the flux
density of electric charge (Section 2.4).)
7. With an expression for the electric field in hand,
we may now compute the potential difference V
between the plates as follows (Section 5.8):
Z
V = − E · dl
C
1
D · (ẑdz)
=−
ǫ
0
Z d
ρs,+ · (ẑdz)
−ẑ
=−
ǫ
0
ρs,+ d
(5.163)
=+
ǫ
Z d
8. Finally,
C=
Q+
ρs,+ A
ǫA
=
=
V
ρs,+ d/ǫ
d
(5.164)
ǫA
d
(5.165)
Summarizing:
C≈
The capacitance of a parallel plate capacitor having plate separation much less than the size of
the plate is given by Equation 5.165. This is an
approximation because the fringing field is neglected.
It’s worth noting that this is dimensionally correct;
i.e., F/m times m2 divided by m yields F. It’s also
worth noting the effect of the various parameters:
Capacitance increases in proportion to permittivity and plate area and decreases in proportion to
distance between the plates.
CHAPTER 5. ELECTROSTATICS
128
Example 5.9. Printed circuit board capacitance.
Printed circuit boards commonly include a
“ground plane,” which serves as the voltage
datum for the board, and at least one “power
plane,” which is used to distribute a DC supply
voltage (See “Additional Reading” at the end of
this section). These planes are separated by a
dielectric material, and the resulting structure
exhibits capacitance. This capacitance may be
viewed as an equivalent discrete capacitor in
parallel with the power supply. The value of this
equivalent capacitor may be either negligible,
significant and beneficial, or significant and
harmful. So, it is useful to know the value of this
equivalent capacitor.
For a common type of circuit board, the
dielectric thickness is about 1.6 mm and the
relative permittivity of the material is about 4.5.
If the area in common between the ground and
power planes is 25 cm2 , what is the value of the
equivalent capacitor?
Solution. From the problem statement,
ǫ∼
= 25 cm2 = 2.5 × 10−3 m2 , and
= 4.5ǫ0 , A ∼
∼
d = 1.6 mm. Using Equation 5.165, the value of
the equivalent capacitor is 62.3 pF.
Additional Reading:
• “Printed circuit board” on Wikipedia.
5.24 Capacitance of a Coaxial
Structure
[m0113]
Let us now determine the capacitance of
coaxially-arranged conductors, shown in Figure 5.15.
Among other applications, this information is useful
in the analysis of voltage and current waves on
coaxial transmission line, as addressed in Sections 3.4
and 3.10.
For our present purposes, we may model the structure
as consisting of two concentric perfectly-conducting
cylinders of radii a and b, separated by an ideal
dielectric having permittivity ǫs . We place the +z
axis along the common axis of the concentric
cylinders so that the cylinders may be described as
constant-coordinate surfaces ρ = a and ρ = b.
In this section, we shall find the capacitance by
assuming a total charge Q+ on the inner conductor
and integrating over the associated electric field to
obtain the voltage between the conductors. Then,
capacitance is computed as the ratio of the assumed
charge to the resulting potential difference. This
strategy is the same as that employed in Section 5.23
for the parallel plate capacitor, so it may be useful to
review that section before attempting this derivation.
The first step is to find the electric field inside the
structure. This is relatively simple if we assume that
b
+
V-
z
a
l
c K. Kikkeri CC BY SA 4.0
Figure 5.15: Determining the capacitance of a coaxial
structure.
5.24. CAPACITANCE OF A COAXIAL STRUCTURE
the structure has infinite length (i.e., l → ∞), since
then there are no fringing fields and the internal field
will be utterly constant with respect to z. In the
central region of a finite-length capacitor, however,
the field is not much different from the field that
exists in the case of infinite length, and if the energy
storage in fringing fields is negligible compared to the
energy storage in this central region then there is no
harm in assuming the internal field is constant with z.
Alternatively, we may think of the length l as
pertaining to one short section of a much longer
structure and thereby obtain the capacitance per
length as opposed to the total capacitance. Note that
the latter is exactly what we need for the transmission
line lumped-element equivalent circuit model
(Section 3.4).
To determine the capacitance, we invoke the
definition (Section 5.22):
C,
Q+
V
(5.166)
where Q+ is the charge on the positively-charged
conductor and V is the potential measured from the
negative conductor to the positive conductor. The
charge on the inner conductor is uniformly-distributed
with density
Q+
ρl =
(5.167)
l
which has units of C/m. Now we will determine the
electric field intensity E, integrate E over a path
between conductors to get V , and then apply
Equation 5.166 to obtain the capacitance.
The electric field intensity for this scenario was
determined in Section 5.6, “Electric Field Due to an
Infinite Line Charge using Gauss’ Law,” where we
found
ρl
(5.168)
E = ρ̂
2πǫs ρ
The reader should note that in that section we were
considering merely a line of charge; not a coaxial
structure. So, on what basis do we claim the field is
the same? This is a consequence of Gauss’ Law
(Section 5.5)
I
D · ds = Qencl
(5.169)
S
which we used in Section 5.6 to find the field. If in
this new problem we specify the same cylindrical
129
surface S with radius ρ < b, then the enclosed charge
is the same. Furthermore, the presence of the outer
conductor does not change the radial symmetry of the
problem, and nothing else remains that can change the
outcome. This is worth noting for future reference:
The electric field inside a coaxial structure comprised of concentric conductors and having uniform charge density on the inner conductor is
identical to the electric field of a line charge in
free space having the same charge density.
Next, we get V using (Section 5.8)
Z
V = − E · dl
(5.170)
C
where C is any path from the negatively-charged outer
conductor to the positively-charged inner conductor.
Since this can be any such path (Section 5.9), we may
as well choose the simplest one. This path is the one
that traverses a radial of constant φ and z. Thus:
Z a ρl
ρ̂
V =−
· (ρ̂dρ)
2πǫs ρ
ρ=b
Z a
dρ
ρl
=−
2πǫs ρ=b ρ
Z b
dρ
ρl
=+
2πǫs ρ=a ρ
ρl
b
=+
(5.171)
ln
2πǫs
a
Wrapping up:
C,
ρl l
Q+
=
V
(ρl /2πǫs ) ln (b/a)
(5.172)
Note that factors of ρl in the numerator and
denominator cancel out, leaving:
C=
2πǫs l
ln (b/a)
(5.173)
Note that this expression is dimensionally correct,
having units of F. Also note that the expression
depends only on materials (through ǫs ) and geometry
(through l, a, and b). The expression does not depend
on charge or voltage, which would imply non-linear
behavior.
CHAPTER 5. ELECTROSTATICS
130
To make the connection back to lumped-element
transmission line model parameters (Sections 3.4 and
3.10), we simply divide by l to get the per-unit length
parameter:
2πǫs
C′ =
(5.174)
ln (b/a)
Example 5.10. Capacitance of RG-59 coaxial
cable.
RG-59 coaxial cable consists of an inner
conductor having radius 0.292 mm, an outer
conductor having radius 1.855 mm, and a
polyethylene spacing material having relative
permittivity 2.25. Estimate the capacitance per
length of RG-59.
Solution. From the problem statement,
a = 0.292 mm, b = 1.855 mm, and ǫs = 2.25ǫ0 .
Using Equation 5.174 we find C ′ = 67.7 pF/m.
5.25 Electrostatic Energy
[m0114]
Consider a structure consisting of two perfect
conductors, both fixed in position and separated by an
ideal dielectric. This could be a capacitor, or it could
be one of a variety of capacitive structures that are not
explicitly intended to be a capacitor – for example, a
printed circuit board. When a potential difference is
applied between the two conducting regions, a
positive charge Q+ will appear on the surface of the
conductor at the higher potential, and a negative
charge Q− = −Q+ will appear on the surface of the
conductor at the lower potential (Section 5.19).
Assuming the conductors are not free to move,
potential energy is stored in the electric field
associated with the surface charges (Section 5.22).
We now ask the question, what is the energy stored in
this field? The answer to this question has relevance
in several engineering applications. For example,
when capacitors are used as batteries, it is useful to
know to amount of energy that can be stored. Also,
any system that includes capacitors or has unintended
capacitance is using some fraction of the energy
delivered by the power supply to charge the
associated structures. In many electronic systems –
and in digital systems in particular – capacitances are
periodically charged and subsequently discharged at a
regular rate. Since power is energy per unit time, this
cyclic charging and discharging of capacitors
consumes power. Therefore, energy storage in
capacitors contributes to the power consumption of
modern electronic systems. We’ll delve into that topic
in more detail in Example 5.11.
Since capacitance C relates the charge Q+ to the
potential difference V between the conductors, this is
the natural place to start. From the definition of
capacitance (Section 5.22):
V =
Q+
C
(5.175)
From Section 5.8, electric potential is defined as the
work done (i.e., energy injected) by moving a charged
particle, per unit of charge; i.e.,
V =
We
q
(5.176)
5.25. ELECTROSTATIC ENERGY
131
where q is the charge borne by the particle and We
(units of J) is the work done by moving this particle
across the potential difference V . Since we are
dealing with charge distributions as opposed to
charged particles, it is useful to express this in terms
of the contribution ∆We made to We by a small
charge ∆q. Letting ∆q approach zero we have
dWe = V dq
(5.177)
Now consider what must happen to transition the
system from having zero charge (q = 0) to the
fully-charged but static condition (q = Q+ ). This
requires moving the differential amount of charge dq
across the potential difference between conductors,
beginning with q = 0 and continuing until q = Q+ .
Therefore, the total amount of work done in this
process is:
Z Q+
dWe
We =
q=0
=
Z Q+
V dq
0
=
Z Q+
0
q
dq
C
1 Q2+
=
2 C
(5.178)
Equation 5.178 can be expressed entirely in terms of
electrical potential by noting again that C = Q+ /V ,
so
1
We = CV 2
(5.179)
2
Since there are no other processes to account for the
injected energy, the energy stored in the electric field
is equal to We . Summarizing:
The energy stored in the electric field of a capacitor (or a capacitive structure) is given by Equation 5.179.
Example 5.11. Why multicore computing is
power-neutral.
Readers are likely aware that computers
increasingly use multicore processors as
opposed to single-core processors. For our
present purposes, a “core” is defined as the
smallest combination of circuitry that performs
independent computation. A multicore processor
consists of multiple identical cores that run in
parallel. Since a multicore processor consists of
N identical processors, you might expect power
consumption to increase by N relative to a
single-core processor. However, this is not the
case. To see why, first realize that the power
consumption of a modern computing core is
dominated by the energy required to
continuously charge and discharge the multitude
of capacitances within the core. From
Equation 5.179, the required energy is 12 C0 V02
per clock cycle, where C0 is the sum capacitance
(remember, capacitors in parallel add) and V0 is
the supply voltage. Power is energy per unit
time, so the power consumption for a single core
is
1
P0 = C0 V02 f0
(5.180)
2
where f0 is the clock frequency. In a N -core
processor, the sum capacitance is increased by
N . However, the frequency is decreased by N
since the same amount of computation is
(nominally) distributed among the N cores.
Therefore, the power consumed by an N -core
processor is
f0
1
= P0
(5.181)
PN = (N C0 ) V02
2
N
In other words, the increase in power associated
with replication of hardware is nominally offset
by the decrease in power enabled by reducing
the clock rate. In yet other words, the total
energy of the N -core processor is N times the
energy of the single core processor at any given
time; however, the multicore processor needs to
recharge capacitances 1/N times as often.
Before moving on, it should be noted that the
usual reason for pursuing a multicore design is
to increase the amount of computation that can
be done; i.e., to increase the product f0 N .
Nevertheless, it is extremely helpful that power
consumption is proportional to f0 only, and is
independent of N .
The thin parallel plate capacitor (Section 5.23) is
representative of a large number of practical
CHAPTER 5. ELECTROSTATICS
132
applications, so it is instructive to consider the
implications of Equation 5.179 for this structure in
particular. For the thin parallel plate capacitor,
C≈
ǫA
d
(5.182)
where A is the plate area, d is the separation between
the plates, and ǫ is the permittivity of the material
between the plates. This is an approximation because
the fringing field is neglected; we shall proceed as if
this is an exact expression. Applying Equation 5.179:
1 ǫA
2
(Ed)
(5.183)
We =
2 d
where E is the magnitude of the electric field intensity
between the plates. Rearranging factors, we obtain:
We =
1 2
ǫE (Ad)
2
(5.184)
Recall that the electric field intensity in the thin
parallel plate capacitor is approximately uniform.
Therefore, the density of energy stored in the
capacitor is also approximately uniform. Noting that
the product Ad is the volume of the capacitor, we find
that the energy density is
we =
We
1
= ǫE 2
Ad
2
(5.185)
which has units of energy per unit volume (J/m3 ).
The above expression provides an alternative method
to compute the total electrostatic energy. Within a
mathematical volume V, the total electrostatic energy
is simply the integral of the energy density over V;
i.e.,
Z
We =
we dv
(5.186)
V
This works even if E and ǫ vary with position. So,
even though we arrived at this result using the
example of the thin parallel-plate capacitor, our
findings at this point apply generally. Substituting
Equation 5.185 we obtain:
We =
Summarizing:
1
2
Z
ǫE 2 dv
V
(5.187)
The energy stored by the electric field present
within a volume is given by Equation 5.187.
It’s worth noting that this energy increases with the
permittivity of the medium, which makes sense since
capacitance is proportional to permittivity.
[m0034]
5.25. ELECTROSTATIC ENERGY
Image Credits
Fig. 5.1: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0102 fCoulombsLaw.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.2: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0104 fRing.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.3: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0104 fDisk.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.4: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:4.5 m0149 commons.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.5: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0064 fResistor.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.7: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0020 fPEC.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.8: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0020 fKVL.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.9: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0021 fPEC.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.10: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0021 fGL.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.11: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0025 fFlatSlab.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.12: by C. Burks, https://commons.wikimedia.org/wiki/File:Electrostatic induction.svg,
in public domain. Minor modifications.
Fig. 5.15: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0113 fCapCoax.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
133
Chapter 6
Steady Current and Conductivity
6.1 Convection and Conduction
Currents
• “Electric current” on Wikipedia.
[m0110]
In practice, we deal with two physical mechanisms
for current: convection and conduction. The
distinction between these types of current is important
in electromagnetic analysis.
Convection current consists of charged particles
moving in response to mechanical forces, as opposed
to being guided by the electric field (Sections 2.2
and/or 5.1). An example of a convection current is a
cloud bearing free electrons that moves through the
atmosphere driven by wind.
Conduction current consists of charged particles
moving in response to the electric field and not
merely being carried by motion of the surrounding
material. In some materials, the electric field is also
able to dislodge weakly-bound electrons from atoms,
which then subsequently travel some distance before
reassociating with other atoms. For this reason, the
individual electrons in a conduction current do not
necessarily travel the full distance over which the
current is perceived to exist.
The distinction between convection and conduction is
important because Ohm’s Law (Section 6.3) – which
specifies the relationship between electric field
intensity and current – applies only to conduction
current.
Additional Reading:
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
6.2. CURRENT DISTRIBUTIONS
6.2 Current Distributions
135
current existing on the wire surface. In this case, the
current is best described as a surface current density
Js , which is the total current I on the wire divided by
the circumference 2πa of the wire:
[m0101]
In elementary electric circuit theory, current is the
rate at which electric charge passes a particular point
in a circuit. For example, 1 A is 1 C per second. In
this view current is a scalar quantity, and there are
only two possible directions because charge is
constrained to flow along defined channels. Direction
is identified by the sign of the current with respect to
a reference direction, which is in turn defined with
respect to a reference voltage polarity. The
convention in electrical engineering defines positive
current as the flow of positive charge into the positive
voltage terminal of a passive device such as a resistor,
capacitor, or inductor. For an active device such as a
battery, positive current corresponds to the flow of
positive charge out of the positive voltage terminal.
However, this kind of thinking only holds up when
the systems being considered are well-described as
“lumped” devices connected by infinitesimally thin,
filament-like connections representing wires and
circuit board traces. Many important problems in
electrical engineering concern situations in which the
flow of current is not limited in this way. Examples
include wire- and pin-type interconnects at radio
frequencies, circuit board and enclosure grounding,
and physical phenomena such as lightning. To
accommodate the more general class of problems, we
must define current as a vector quantity. Furthermore,
current in these problems can spread out over surfaces
and within volumes, so we must also consider spatial
distributions of current.
Line Current. As noted above, if a current I is
constrained to follow a particular path, then the only
other consideration is direction. Thus, a line current is
specified mathematically as l̂I, where the direction l̂
may vary with position along the path. For example,
in a straight wire l̂ is constant, whereas in a coil l̂
varies with position along the coil.
Surface Current Distribution. In some cases,
current may be distributed over a surface. For
example, the radio-frequency current on a wire of
radius a made from a metal with sufficiently high
conductivity can be modeled as a uniform surface
Js = û
I
(units of A/m)
2πa
(6.1)
where û is the direction of current flow.
Volume Current Distribution. Imagine that current
is distributed within a volume. Let û∆i be the current
passing through a small open planar surface defined
within this volume, and let ∆s be the area of this
surface. The volume current density J at any point in
the volume is defined as
J , lim
∆s→0
û∆i
di
= û
(units of A/m2 )
∆s
ds
(6.2)
In general, J is a function of position within this
volume. Subsequently the total current passing
through a surface S is
Z
J · ds (units of A)
(6.3)
I=
S
In other words, volume current density integrated over
a surface yields total current through that surface. You
might recognize this as a calculation of flux, and it is.1
Example 6.1. Current and current density in a
wire of circular cross-section.
Figure 6.1 shows a straight wire having
cross-sectional radius a = 3 mm. A battery is
connected across the two ends of the wire
resulting in a volume current density
J = ẑ8 A/m2 , which is uniform throughout the
wire. Find the net current I through the wire.
Solution. The net current is
Z
I=
J · ds
S
We choose S to be the cross-section
perpendicular to the axis of the wire. Also, we
choose ds to point such that I is positive with
respect to the sign convention shown in
Figure 6.1 , which is the usual choice in electric
1 However, it is not common to refer to net current as a flux; go
figure!
CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY
136
6.3 Conductivity
a
z
[m0010]
J
ds
Conductivity is one of the three primary “constitutive
parameters” that is commonly used to characterize the
electromagnetic properties of materials (Section 2.8).
The key idea is this:
I
+ c K. Kikkeri CC BY SA 4.0
Figure 6.1: Net current I and current density J in a
wire of circular cross-section.
circuit analysis. With these choices, we have
I=
Z a Z 2π
ρ=0
φ=0
= 8 A/m
2
ẑ8 A/m2 · (ẑρ dρ dφ)
Z a Z 2π
ρ=0
ρ dρ dφ
φ=0
= 8 A/m2 · πa2
= 226 µA
This answer is independent of the
cross-sectional surface S used to do the
calculation. For example, you could use an
alternative surface that is tilted 45◦ away from
the axis of the wire. In that case, the
cross-sectional area would increase, but the dot
product of J and ds would be proportionally less
and the outcome would be the same. Since the
choice of S is arbitrary – any surface with edges
at the perimeter of the wire will do – you should
make the choice that makes the problem as
simple as possible, as we have done above.
Additional Reading:
• “Electric current” on Wikipedia.
• “Current density” on Wikipedia.
Conductivity is a property of materials that determines conduction current density in response to
an applied electric field.
Recall that conduction current is the flow of charge in
response to an electric field (Section 6.1). Although
the associated force is straightforward to calculate
(e.g., Section 5.1), the result is merely the force
applied, not the speed at which charge moves in
response. The latter is determined by the mobility of
charge, which is in turn determined by the atomic and
molecular structure of the material. Conductivity
relates current density to the applied field directly,
without requiring one to grapple separately with the
issues of applied force and charge mobility.
In the absence of material – that is, in a true, perfect
vacuum – conductivity is zero because there is no
charge available to form current, and therefore the
current is zero no matter what electric field is applied.
At the other extreme, a good conductor is a material
that contains a supply of charge that is able to move
freely within the material. When an electric field is
applied to a good conductor, charge-bearing material
constituents move in the direction determined by the
electric field, creating current flow in that direction.
This relationship is summarized by Ohm’s Law for
Electromagnetics:
J = σE
(6.4)
where E is electric field intensity (V/m); J is the
volume current density, a vector describing the
current flow, having units of A/m2 (see Section 6.2);
and σ is conductivity. Since E has units of V/m, we
see σ has units of Ω−1 m−1 , which is more commonly
expressed as S/m, where 1 S (“siemens”) is defined as
1 Ω−1 . Section A.3 provides values of conductivity
for a representative set of materials.
6.3. CONDUCTIVITY
Conductivity σ is expressed in units of S/m,
where 1 S = 1 Ω−1 .
It is important to note that the current being addressed
here is conduction current, and not convection
current, displacement current, or some other form of
current – see Section 6.2 for elaboration.
Summarizing:
Ohm’s Law for Electromagnetics (Equation 6.4)
states that volume density of conduction current
(A/m2 ) equals conductivity (S/m) times electric
field intensity (V/m).
The reader is likely aware that there is also an “Ohm’s
Law” in electric circuit theory, which states that
current I (units of A) is voltage V (units of V)
divided by resistance R (units of Ω); i.e., I = V /R.
This is in fact a special case of Equation 6.4; see
Section 6.4 for more about this.
As mentioned above, σ depends on both the
availability and mobility of charge within the
material. At the two extremes, we have perfect
insulators, for which σ = 0, and perfect conductors,
for which σ → ∞. Some materials approach these
extremes, whereas others fall midway between these
conditions. Here are a few classes of materials that
are frequently encountered:
• A perfect vacuum – “free space” – contains no
charge and therefore is a perfect insulator with
σ = 0.
• Good insulators typically have conductivities
≪ 10−10 S/m, which is sufficiently low that the
resulting currents can usually be ignored. The
most important example is air, which has a
conductivity only slightly greater than that of
free space. An important class of good insulators
is lossless dielectrics,2 which are
well-characterized in terms of permittivity (ǫ)
alone, and for which µ = µ0 and σ = 0 may
usually be assumed.
• Poor insulators have conductivities that are low,
but nevertheless sufficiently high that the
2 See Section 5.20 for a discussion of dielectric materials.
137
resulting currents cannot be ignored. For
example, the dielectric material that is used to
separate the conductors in a transmission line
must be considered a poor insulator as opposed
to a good (effectively lossless) insulator in order
to characterize loss per length along the
transmission line, which can be significant.3
These lossy dielectrics are well-characterized in
terms of ǫ and σ, and typically µ = µ0 can be
assumed.
• Semiconductors such as those materials used in
integrated circuits have intermediate
conductivities, typically in the range 10−4 to
10+1 S/m.
• Good conductors are materials with very high
conductivities, typically greater than 105 S/m.
An important category of good conductors
includes metals, with certain metals including
alloys of aluminum, copper, and gold reaching
conductivities on the order of 108 S/m. In such
materials, minuscule electric fields give rise to
large currents. There is no significant storage of
energy in such materials, and so the concept of
permittivity is not relevant for good conductors.
The reader should take care to note that terms such as
good conductor and poor insulator are qualitative and
subject to context. What may be considered a “good
insulator” in one application may be considered to be
a “poor insulator” in another.
One relevant category of material was not included in
the above list – namely, perfect conductors. A perfect
conductor is a material in which σ → ∞. It is
tempting to interpret this as meaning that any electric
field gives rise to infinite current density; however,
this is not plausible even in the ideal limit. Instead,
this condition is interpreted as meaning that
E = J/σ → 0 throughout the material; i.e., E is zero
independently of any current flow in the material. An
important consequence is that the potential field V is
equal to a constant value throughout the material.
(Recall E = −∇V (Section 5.14), so constant V
means E = 0.) We refer to the volume occupied by
such a material as an equipotential volume. This
concept is useful as an approximation of the behavior
3 Review Sections 3.4, 3.9, and associated sections for a refresher
on this issue.
138
CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY
of good conductors; for example, metals are often
modeled as perfectly-conducting equipotential
volumes in order to simplify analysis.
6.4 Resistance
[m0071]
A perfect conductor is a material for which σ →
∞, E → 0, and subsequently V (the electric potential) is constant.
One final note: It is important to remain aware of the
assumptions we have made about materials in this
book, which are summarized in Section 2.8. In
particular, we continue to assume that materials are
linear unless otherwise indicated. For example,
whereas air is normally considered to be a good
insulator and therefore a poor conductor, anyone who
has ever witnessed lightning has seen a demonstration
that under the right conditions – i.e., sufficiently large
potential difference between earth and sky – current
will readily flow through air. This particular situation
is known as dielectric breakdown (see Section 5.21).
The non-linearity of materials can become evident
before reaching the point of dielectric breakdown, so
one must be careful to consider this possibility when
dealing with strong electric fields.
Additional Reading:
• “Electrical Resistivity and Conductivity” on
Wikipedia.
• “Ohm’s Law” on Wikipedia.
The concept of resistance is most likely familiar to
readers via Ohm’s Law for Devices; i.e., V = IR
where V is the potential difference associated with a
current I. This is correct, but it is not the whole story.
Let’s begin with a statement of intent:
Resistance R (Ω) is a characterization of the conductivity of a device (as opposed to a material) in
terms of Ohms Law for Devices; i.e., V = IR.
Resistance is a property of devices such as resistors,
which are intended to provide resistance, as well as
being a property of most practical electronic devices,
whether it is desired or not.
Resistance is a manifestation of the conductivity of
the materials comprising the device, which
subsequently leads to the “V = IR” relationship.
This brings us to a very important point and a
common source of confusion. Resistance is not
necessarily the real part of impedance. Let’s take a
moment to elaborate. Impedance (Z) is defined as the
ratio of voltage to current; i.e., V /I; or equivalently in
e Most devices – not just
the phasor domain as Ve /I.
devices exhibiting resistance – can be characterized in
terms of this ratio. Consider for example the input
impedance of a terminated transmission line
(Section 3.15). This impedance may have a non-zero
real-valued component even when the transmission
line and the terminating load are comprised of perfect
conductors. Summarizing:
Resistance results in a real-valued impedance.
However, not all devices exhibiting a real-valued
impedance exhibit resistance. Furthermore, the
real component of a complex-valued impedance
does not necessarily represent resistance.
Restating the main point in yet other words:
Resistance pertains to limited conductivity, not simply
to voltage-current ratio.
Also important to realize is that whereas conductivity
σ (units of S/m) is a property of materials, resistance
depends on both conductivity and the geometry of the
6.4. RESISTANCE
139
z=0
z=
a
σ
z
E
We do not yet know E; however, we know it is
constant throughout the device and points in the −ẑ
direction since this is the direction of current flow and
Ohm’s Law for Electromagnetics (Section 6.3)
requires the electric field to point in the same
direction. Thus, we may write E = −ẑEz where Ez
is a constant. We now find:
Z l
(−ẑEz ) · (ẑdz)
V =−
z=0
-
+
= +Ez
I
V
Z l
dz
z=0
= +Ez l
c K. Kikkeri CC BY SA 4.0
Figure 6.2: Analysis of the resistance of straight wire The current I is given by (Section 6.2)
Z
of circular cross-section.
I=
J · ds
device. In this section, we address the question of
how the resistance of a device can be is determined.
The following example serves this purpose.
Figure 6.2 shows a straight wire of length l centered
on the z axis, forming a cylinder of material having
conductivity σ and cross-sectional radius a. The ends
of the cylinder are covered by perfectly-conducting
plates to which the terminals are attached. A battery
is connected to the terminals, resulting in a uniform
internal electric field E between the plates.
To calculate R for this device, let us first calculate V ,
then I, and finally R = V /I. First, we compute the
potential difference using the following result from
Section 5.8:
Z
(6.5)
V = − E · dl
S
We choose the surface S to be the cross-section
perpendicular to the axis of the wire. Also, we choose
ds to point such that I is positive with respect to the
sign convention shown in Figure 6.2. With these
choices, we have
Z a Z 2π
J · (−ẑ ρ dρ dφ)
I=
ρ=0
From Ohm’s Law for Electromagnetics, we have
J = σE = −ẑσEz
z=0
(6.6)
So now
I=
Z a Z 2π
= σEz
(−ẑσEz ) · (−ẑρ dρ dφ)
φ=0
Z a Z 2π
ρ=0
C
Here we can view V as a “given” (being the voltage
of the battery), but wish to evaluate the right hand
side so as to learn something about the effect of the
conductivity and geometry of the wire. The
appropriate choice of C begins at z = 0 and ends at
z = l, following the axis of the cylinder. (Remember:
C defines the reference direction for increasing
potential, so the resulting potential difference will be
the node voltage at the end point minus the node
voltage at the start point.) Thus, dl = ẑdz and we
have
Z l
E · (ẑdz)
V =−
φ=0
ρ dρ dφ
ρ=0 φ=0
2
= σEz πa
Finally:
R=
V
Ez l
l
=
=
I
σEz (πa2 )
σ (πa2 )
This is a good-enough answer for the problem posed,
but it is easily generalized a bit further. Noting that
πa2 in the denominator is the cross-sectional area A
of the wire, so we find:
R=
l
σA
(6.7)
140
CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY
i.e., the resistance of a wire having cross-sectional
area A – regardless of the shape of the cross-section,
is given by the above equation.
The resistance of a right cylinder of material,
given by Equation 6.7, is proportional to length
and inversely proportional to cross-sectional area
and conductivity.
Example 6.3. Resistance of steel pipe.
A pipe is 3 m long and has inner and outer radii
of 5 mm and 7 mm respectively. It is made from
steel having conductivity 4 MS/m. What is the
DC resistance of this pipe?
Solution. We can use Equation 6.7 if we can
determine the cross-sectional area A through
which the current flows. This area is simply the
area defined by the outer radius, πb2 , minus the
area defined by the inner radius πa2 . Thus,
A = πb2 − πa2 ∼
= 7.54 × 10−5 m2 . From the
problem statement, we also determine that
σ = 4 × 106 S/m and l = 3 m. Using
Equation 6.7 we obtain R ∼
= 9.95 mΩ.
It is important to remember that Equation 6.7
presumes that the volume current density J is uniform
over the cross-section of the wire. This is an excellent
approximation for thin wires at “low” frequencies
including, of course, DC. At higher frequencies it
may not be a good assumption that J is
uniformly-distributed over the cross-section of the
wire, and at sufficiently high frequencies one finds
instead that the current is effectively limited to the
Additional Reading:
exterior surface of the wire. In the “high frequency”
case, A in Equation 6.7 is reduced from the physical
• “Resistor” on Wikipedia.
area to a smaller value corresponding to the reduced
area through which most of the current flows.
• “Ohm’s Law” on Wikipedia.
Therefore, R increases with increasing frequency. To
• “American wire gauge” on Wikipedia.
quantify the high frequency behavior of R (including
determination of what constitutes “high frequency” in
this context) one requires concepts beyond the theory
of electrostatics, so this is addressed elsewhere.
Example 6.2. Resistance of 22 AWG hookup
wire.
A common type of wire found in DC
applications is 22AWG (“American Wire
Gauge”; see “Additional Resources” at the end
of this section) copper solid-conductor hookup
wire. This type of wire has circular cross-section
with diameter 0.644 mm. What is the resistance
of 3 m of this wire? Assume copper conductivity
of 58 MS/m.
Solution. From the problem statement, the
diameter 2a = 0.644 mm, σ = 58 × 106 S/m,
and l = 3 m. The cross-sectional area is
A = πa2 ∼
= 3.26 × 10−7 m2 . Using
Equation 6.7 we obtain R = 159 mΩ.
6.5. CONDUCTANCE
6.5 Conductance
[m0105]
Conductance, like resistance (Section 6.4), is a
property of devices. Specifically:
Conductance G (Ω−1 or S) is the reciprocal of
resistance R.
Therefore, conductance depends on both the
conductivity of the materials used in the device, as
well as the geometry of the device.
A natural question to ask is, why do we require the
concept of conductance, if it simply the reciprocal of
resistance? The short answer is that the concept of
conductance is not required; or, rather, we need only
resistance or conductance and not both. Nevertheless,
the concept appears in engineering analysis for two
reasons:
• Conductance is sometimes considered to be a
more intuitive description of the underlying
physics in cases where the applied voltage is
considered to be the independent “stimulus” and
current is considered to be the response. This is
why conductance appears in the lumped element
model for transmission lines (Section 3.4), for
example.
• Characterization in terms of conductance may be
preferred when considering the behavior of
devices in parallel, since the conductance of a
parallel combination is simply the sum of the
conductances of the devices.
Example 6.4. Conductance of a coaxial
structure.
Let us now determine the conductance of a
structure consisting of coaxially-arranged
conductors separated by a lossy dielectric, as
shown in Figure 6.3. The conductance per unit
length G′ (i.e., S/m) of this structure is of
interest in determining the characteristic
impedance of coaxial transmission line, as
addressed in Sections 3.4 and 3.10.
141
For our present purposes, we may model the
structure as two concentric perfectly-conducting
cylinders of radii a and b, separated by a lossy
dielectric having conductivity σs . We place the
+z axis along the common axis of the concentric
cylinders so that the cylinders may be described
as constant-coordinate surfaces ρ = a and ρ = b.
There are at least 2 ways to solve this problem.
One method is to follow the procedure that was
used to find the capacitance of this structure in
Section 5.24. Adapting that approach to the
present problem, one would assume a potential
difference V between the conductors, from that
determine the resulting electric field intensity E,
and then using Ohm’s Law for Electromagnetics
(Section 6.3) determine the density J = σs E of
the current that leaks directly between
conductors. From this, one is able to determine
the total leakage current I, and subsequently the
conductance G , I/V . Although highly
recommended as an exercise for the student, in
this section we take an alternative approach so as
to demonstrate that there are a variety of
approaches available for such problems.
The method we shall use below is as follows: (1)
Assume a leakage current I between the
conductors; (2) Determine the associated current
density J, which is possible using only
geometrical considerations; (3) Determine the
associated electric field intensity E using J/σs ;
(4) Integrate E over a path between the
conductors to get V . Then, as before,
conductance G , I/V .
The current I is defined as shown in Figure 6.3,
with reference direction according to the
engineering convention that positive current
flows out of the positive terminal of a source.
The associated current density must flow in the
same direction, and the circular symmetry of the
problem therefore constrains J to have the form
J = ρ̂
I
A
(6.8)
where A is the area through which I flows. In
other words, current flows radially outward from
the inner conductor to the outer conductor, with
CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY
142
ρ
density that diminishes inversely with the area
through which the total current flows. (It may be
helpful to view J as a flux density and I as a
flux, as noted in Section 6.2.) This area is simply
circumference 2πρ times length l, so
J = ρ̂
I
2πρl
(6.9)
which exhibits the correct units of A/m2 .
J
I
= ρ̂
σs
2πρlσs
Next we get V using (Section 5.8)
Z
V = − E · dl
(6.10)
(6.11)
C
where C is any path from the negatively-charged
outer conductor to the positively-charged inner
conductor. Since this can be any such path
(Section 5.9), we should choose the simplest
one. The simplest path is the one that traverses a
radial of constant φ and z. Thus:
Z a I
ρ̂
· (ρ̂dρ)
V =−
2πρlσs
ρ=b
Z a
dρ
I
=−
2πlσs ρ=b ρ
Z b
dρ
I
=+
2πlσs ρ=a ρ
I
b
=+
(6.12)
ln
2πlσs
a
Wrapping up:
G,
I
I
=
V
(I/2πlσs ) ln (b/a)
(6.13)
Note that factors of I in the numerator and
denominator cancel out, leaving:
G=
2πlσs
ln (b/a)
b
z
I
J
a
l
Now from Ohm’s Law for Electromagnetics we
find the electric field within the structure is
E=
s
(6.14)
Figure 6.3: Determining the conductance of a structure consisting of coaxially-arranged conductors separated by a lossy dielectric.
Note that Equation 6.14 is dimensionally
correct, having units of S = Ω−1 . Also note that
this is expression depends only on materials
(through σs ) and geometry (through l, a, and b).
Notably it does not depend on current or voltage,
which would imply non-linear behavior.
To make the connection back to lumped-element
transmission line model parameters
(Sections 3.4 and 3.10), we simply divide by l to
get the per-length parameter:
G′ =
2πσs
ln (b/a)
(6.15)
Example 6.5. Conductance of RG-59 coaxial
cable.
RG-59 coaxial cable consists of an inner
conductor having radius 0.292 mm, an outer
conductor having radius 1.855 mm, and a
polyethylene spacing material exhibiting
conductivity of about 5.9 × 10−5 S/m. Estimate
the conductance per length of RG-59.
Solution. From the problem statement,
a = 0.292 mm, b = 1.855 mm, and
σs ∼
= 5.9 × 10−5 S/m. Using Equation 6.15, we
find G′ ∼
= 200 µS/m.
6.6. POWER DISSIPATION IN CONDUCTING MEDIA
6.6 Power Dissipation in
Conducting Media
[m0106]
The displacement of charge in response to the force
exerted by an electric field constitutes a reduction in
the potential energy of the system (Section 5.8). If the
charge is part of a steady current, there must be an
associated loss of energy that occurs at a steady rate.
Since power is energy per unit time, the loss of energy
associated with current is expressible as power
dissipation. In this section, we address two questions:
(1) How much power is dissipated in this manner, and
(2) What happens to the lost energy?
First, recall that work is force times distance traversed
in response to that force (Section 5.8). Stated
mathematically:
∆W = +F · ∆l
The associated power ∆P (units of W) is ∆W
divided by the time ∆t (units of s) required for the
distance ∆l to be traversed:
∆l
∆W
=F·
∆t
∆t
particle bearing charge q (units of C) is qE
(Section 2.2). However, we’d like to express this
force in terms of a current, as opposed to a charge. An
expression in terms of current can be constructed as
follows. First, note that the total charge in a small
volume “cell” is the volume charge density ρv (units
of C/m3 ) times the volume ∆v of the cell; i.e.,
q = ρv ∆v (Section 5.3). Therefore:
F = qE = ρv ∆v E
∆P = ρv ∆v E ·
4 It could be argued that it is bad form to use the same variable
to represent both tallies; nevertheless, it is common practice and so
we simply remind the reader that it is important to be aware of the
definitions of variables each time they are (re)introduced.
∆l
∆l
∆v
= E · ρv
∆t
∆t
(6.19)
The quantity in parentheses has units of C/m3 · m ·
s−1 , which is A/m2 . Apparently this quantity is the
volume current density J, so we have
∆P = E · J ∆v
(6.20)
In the limit as ∆v → 0 we have
dP = E · J dv
(6.21)
and integrating over the volume V of interest we
obtain
Z
Z
P =
dP =
V
V
E · J dv
(6.22)
The above expression is commonly known as Joule’s
Law. In our situation, it is convenient to use Ohm’s
Law for Electromagnetics (J = σE; Section 6.3) to
get everything in terms of materials properties (σ),
geometry (V), and the electric field:
Z
P =
E · (σE) dv
(6.23)
V
which is simply
(6.17)
Now we’d like to express force in terms of the electric
field exerting this force. Recall that the force exerted
by an electric field intensity E (units of V/m) on a
(6.18)
and subsequently
(6.16)
where the vector F is the force (units of N) exerted by
the electric field, the vector ∆l is the direction and
distance (units of m) traversed, and ∆W is the work
done (units of J) as a result. Note that a “+” has been
explicitly indicated; this is to emphasize the
distinction from the work being considered in
Section 5.8. In that section, the work “∆W ”
represented energy from an external source that was
being used to increase the potential energy of the
system by moving charge “upstream” relative to the
electric field. Now, ∆W represents this internal
energy as it is escaping from the system in the form of
kinetic energy; therefore, positive ∆W now means a
reduction in potential energy, hence the sign change.4
∆P =
143
P =
Z
2
V
σ |E| dv
(6.24)
Thus:
The power dissipation associated with current
is given by Equation 6.24. This power is proportional to conductivity and proportional to the
electric field magnitude squared.
CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY
144
This result facilitates the analysis of power dissipation
in materials exhibiting loss; i.e., having finite
conductivity. But what is the power dissipation in a
perfectly conducting material? For such a material,
σ → ∞ and E → 0 no matter how much current is
applied (Section 6.3). In this case, Equation 6.24 is
not very helpful. However, as we just noted, being a
perfect conductor means E → 0 no matter how much
current is applied, so from Equation 6.22 we have
found that:
No power is dissipated in a perfect conductor.
When conductivity is finite, Equation 6.24 serves as a
more-general version of a concept from elementary
circuit theory, as we shall now demonstrate. Let
2
E = ẑEz , so |E| = Ez2 . Then Equation 6.24
becomes:
Z
Z
P =
σEz2 dv = σEz2
dv
(6.25)
V
V
The second integral in Equation 6.25 is a calculation
of volume. Let’s assume V is a cylinder aligned along
the z axis. The volume of this cylinder is the
cross-sectional area A times the length l. Then the
above equation becomes:
P = σEz2 A l
(6.26)
For reasons that will become apparent very shortly,
let’s reorganize the above expression as follows:
P = (σEz A) (Ez l)
(6.27)
Note that σEz is the current density in A/m2 , which
when multiplied by A gives the total current.
Therefore, the quantity in the first set of parentheses
is simply the current I. Also note that Ez l is the
potential difference over the length l, which is simply
the node-to-node voltage V (Section 5.8). Therefore,
we have found:
P = IV
(6.28)
as expected from elementary circuit theory.
Now, what happens to the energy associated with this
dissipation of power? The displacement of charge
carriers in the material is limited by the conductivity,
which itself is finite because, simply put, other
constituents of the material get in the way. If charge is
being displaced as described in this section, then
energy is being used to displace the charge-bearing
particles. The motion of constituent particles is
observed as heat – in fact, this is essentially the
definition of heat. Therefore:
The power dissipation associated with the flow of
current in any material that is not a perfect conductor manifests as heat.
This phenomenon is known as joule heating, ohmic
heating, and by other names. This conversion of
electrical energy to heat is the method of operation for
toasters, electric space heaters, and many other
devices that generate heat. It is of course also the
reason that all practical electronic devices generate
heat.
Additional Reading:
• “Joule Heating” on Wikipedia.
[m0057]
6.6. POWER DISSIPATION IN CONDUCTING MEDIA
Image Credits
Fig. 6.1: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0101 fMatEx1.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 6.2: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0071 fMatEx1.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
145
Chapter 7
Magnetostatics
[m0117]
Magnetostatics is the theory of the magnetic field in
conditions in which its behavior is independent of
electric fields, including
• The magnetic field associated with various
spatial distributions of steady current
The technical term for these similarities is duality.
Duality also exists between voltage and current in
electrical circuit theory. For more about the concept
of duality, see “Additional Reading” at the end of this
section.
Additional Reading:
• The energy associated with the magnetic field
• “Duality (electricity and magnetism)” on
Wikipedia.
• Inductance, which is the ability of a structure to
store energy in a magnetic field
• “Duality (electrical circuits)” on Wikipedia.
The word ending “-statics” refers to the fact that these
aspects of electromagnetic theory can be developed
by assuming that the sources of the magnetic field are
time-invariant; we might say that magnetostatics is
the study of the magnetic field at DC. However, many
aspects of magnetostatics are applicable at “AC” as
well.
7.1 Comparison of Electrostatics
and Magnetostatics
[m0115]
Students encountering magnetostatics for the first
time have usually been exposed to electrostatics
already. Electrostatics and magnetostatics exhibit
many similarities. These are summarized in Table 7.1.
The elements of magnetostatics presented in this table
are all formally introduced in other sections; the sole
purpose of this table is to point out the similarities.
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
7.2. GAUSS’ LAW FOR MAGNETIC FIELDS: INTEGRAL FORM
Sources
Field intensity
Flux density
Material relations
Force on charge q
Maxwell’s Eqs.
(integral)
Maxwell’s Eqs.
(differential)
Boundary Conditions
Energy storage
Energy density
Energy dissipation
electrostatics
static charge
E (V/m)
D (C/m2 )
D = ǫE
J = σE
F
H = qE
HS D · ds = Qencl
E · dl = 0
C
∇ · D = ρv
∇×E=0
n̂ × [E1 − E2 ] = 0
n̂ · [D1 − D2 ] = ρs
Capacitance
we = 12 ǫE 2
Resistance
147
magnetostatics
steady current,
magnetizable material
H (A/m)
B (Wb/m2 =T)
B = µH
F
H = qv × B
HS B · ds = 0
H · dl = Iencl
C
∇·B=0
∇×H=J
n̂ × [H1 − H2 ] = Js
n̂ · [B1 − B2 ] = 0
Inductance
wm = 12 µH 2
Table 7.1: A summary of the duality between electrostatics and magnetostatics.
7.2 Gauss’ Law for Magnetic
Fields: Integral Form
surface gives a quantity with units of Wb, which is
magnetic flux, as indicated above.
[m0018]
Gauss’ Law for Magnetic Fields (GLM) is one of the
four fundamental laws of classical electromagnetics,
collectively known as Maxwell’s Equations. Before
diving in, the reader is strongly encouraged to review
Section 2.5. In that section, GLM emerges from the
“flux density” interpretation of the magnetic field.
GLM is not identified in that section, but now we are
ready for an explicit statement:
Gauss’ Law for Magnetic Fields (Equation 7.1)
states that the flux of the magnetic field through
a closed surface is zero.
This is expressed mathematically as follows:
I
S
B · ds = 0
(7.1)
where B is magnetic flux density and S is a closed
surface with outward-pointing differential surface
normal ds. It may be useful to consider the units. B
has units of Wb/m2 ; therefore, integrating B over a
GLM can also be interpreted in terms of magnetic
field lines. For the magnetic flux through a closed
surface to be zero, every field line entering the
volume enclosed by S must also exit this volume –
field lines may not begin or end within the volume.
The only way this can be true for every possible
surface S is if magnetic field lines always form closed
loops. This is in fact what we find in practice, as
shown in Figure 7.1.
Following this argument one step further, GLM
implies there can be no particular particle or structure
that can be the source of the magnetic field (because
then that would be a start point for field lines). This is
one way in which the magnetic field is very different
from the electrostatic field, for which every field line
begins at a charged particle. So, when we say that
current (for example) is the source of the magnetic
field, we mean only that the field coexists with
current, and not that the magnetic field is somehow
attached to the current. Summarizing, there is no
“localizable” quantity, analogous to charge for
electric fields, associated with magnetic fields. This is
just another way in which magnetic fields are weird!
CHAPTER 7. MAGNETOSTATICS
148
7.3 Gauss’ Law for Magnetism:
Differential Form
[m0047]
The integral form of Gauss’ Law (Section 7.2) states
that the magnetic flux through a closed surface is
zero. In mathematical form:
I
B · ds = 0
(7.2)
S
where B is magnetic flux density and S is the
enclosing surface. Just as Gauss’s Law for
electrostatics has both integral (Sections 5.5) and
differential (Section 5.7) forms, so too does Gauss’
c Youming / K. Kikkeri CC BY SA 4.0
Law for Magnetic Fields. Here we are interested in
the differential form for the same reason. Given a
Figure 7.1: Gauss’ Law for Magnetostatics applied to
differential equation and the boundary conditions
a two-dimensional bar magnet. For the surface S =
imposed by structure and materials, we may then
SA , every field line entering S also leaves S, so the
solve for the magnetic field in very complicated
flux through S is zero. For the surface S = SB , every
scenarios.
field line within S remains in S, so the flux through S
is again zero.
The equation we seek may be obtained from
Summarizing:
Gauss’ Law for Magnetic Fields requires that
magnetic field lines form closed loops. Furthermore, there is no particle that can be identified as
the source of the magnetic field.
Additional Reading:
• “Gauss’ Law for Magnetism” on Wikipedia.
• “Maxwell’s Equations” on Wikipedia.
Equation 7.2 using the Divergence Theorem
(Section 4.7), which in the present case may be
written:
Z
I
V
(∇ · B) dv =
S
B · ds
(7.3)
Where V is the mathematical volume bounded by the
closed surface S. From Equation 7.2 we see that the
right hand side of the equation is zero, leaving:
Z
(∇ · B) dv = 0
(7.4)
V
The above relationship must hold regardless of the
specific location or shape of V. The only way this is
possible is if the integrand is everywhere equal to
zero. We conclude:
∇·B=0
(7.5)
The differential (“point”) form of Gauss’ Law
for Magnetic Fields (Equation 7.5) states that the
flux per unit volume of the magnetic field is always zero.
7.4. AMPERE’S CIRCUITAL LAW (MAGNETOSTATICS): INTEGRAL FORM
This is another way of saying that there is no point in
space that can be considered to be the source of the
magnetic field, for if it were, then the total flux
through a bounding surface would be greater than
zero. Said yet another way, the source of the magnetic
field is not localizable.
149
7.4 Ampere’s Circuital Law
(Magnetostatics): Integral
Form
[m0019]
Additional Reading:
• “Gauss’ Law for Magnetism” on Wikipedia.
Ampere’s circuital law (ACL) relates current to the
magnetic field associated with the current. In the
magnetostatic regime, the law is (see also Figure 7.2):
I
C
H · dl = Iencl
(7.6)
That is, the integral of the magnetic field intensity H
over a closed path C is equal to the current enclosed
by that path, Iencl . Before proceeding to interpret this
law, it is useful to see that it is dimensionally correct.
That is, H (units of A/m) integrated over a distance
(units of m) yields a quantity with units of current
(i.e., A).
In general, Iencl may be either positive or negative.
The direction corresponding to positive current flow
must be correctly associated with C. The relationship
follows the right-hand rule of Stokes’ Theorem
(Section 4.9) summarized as follows. The direction of
positive Iencl is the direction in which the fingers of
the right hand intersect any surface S bordered by C
when the thumb of the right hand points in the
direction of integration. The connection to Stokes’
Theorem is not a coincidence. See Section 7.9 for
more about this.
Iencl
c K. Kikkeri CC BY SA 4.0
Figure 7.2: Reference directions for Ampere’s Circuital Law (Equation 7.6).
CHAPTER 7. MAGNETOSTATICS
150
Note that S can be any surface that is bounded by C –
not just the taut surface implied in Figure 7.2.
The integral form of Ampere’s Circuital Law for
magnetostatics (Equation 7.6) relates the magnetic field along a closed path to the total current flowing through any surface bounded by that
path.
ACL plays a role in magnetostatics that is very
similar to the role played by the integral form of
Gauss’ Law in electrostatics (Section 5.5). That is,
ACL provides a means calculate the magnetic field
given the source current. ACL also has a similar
limitation. Generally, symmetry is required to
simplify the problem sufficiently so that the integral
equation may be solved. Fortunately, a number of
important problems fall in this category. Examples
include the problems addressed in Sections 7.5
(magnetic field of a line current), 7.6 (magnetic field
inside a straight coil), 7.7 (magnetic field of a toroidal
coil), 7.8 (magnetic field of a current sheet), and 7.11
(boundary conditions on the magnetic field intensity).
For problems in which the necessary symmetry is not
available, the differential form of ACL may be
required (Section 7.9).
7.5 Magnetic Field of an
Infinitely-Long Straight
Current-Bearing Wire
[m0119]
In this section, we use the magnetostatic form of
Ampere’s Circuital Law (ACL) (Section 7.4) to
determine the magnetic field due to a steady current I
(units of A) in an infinitely-long straight wire. The
problem is illustrated in Figure 7.3. The wire is an
electrically-conducting circular cylinder of radius a.
Since the wire is a cylinder, the problem is easiest to
work in cylindrical coordinates with the wire aligned
along the z axis.
Here’s the relevant form of ACL:
I
H · dl = Iencl
(7.7)
C
where Iencl is the current enclosed by the closed path
C. ACL works for any closed path, so to exploit the
z
Finally, be aware that the form of ACL addressed here
applies to magnetostatics only. In the presence of a
time-varying electric field, the right side of ACL
includes an additional term known as the
displacement current (Section 8.9).
y
Additional Reading:
ρ
• “Maxwell’s Equations” on Wikipedia.
x
• “Ampere’s Circuital Law” on Wikipedia.
a
I
c K. Kikkeri CC BY SA 4.0
Figure 7.3: Determination of the magnetic field due
to steady current in an infinitely-long straight wire.
7.5. MAGNETIC FIELD OF AN INFINITELY-LONG STRAIGHT CURRENT-BEARING WIRE
symmetry of the cylindrical coordinate system we
choose a circular path of radius ρ in the z = 0 plane,
centered at the origin. With this choice we have
Iencl = I for ρ ≥ a
(7.8)
For ρ < a, we see that Iencl < I. a steady (DC)
current will be distributed uniformly throughout the
wire (Section 6.4). Since the current is uniformly
distributed over the cross section, Iencl is less than the
total current I by the same factor that the area
enclosed by C is less than πa2 , the cross-sectional
area of the wire. The area enclosed by C is simply
πρ2 , so we have
Iencl = I
πρ2
ρ2
= I 2 for ρ < a
2
πa
a
151
• The radial symmetry of the problem also
requires that H · ρ̂ be equal to zero. If this were
not the case, then the field would not be radially
symmetric. Since we determined above that H · ẑ
is also zero, H must be entirely ±φ̂-directed.
From the above considerations, the most general form
of the magnetic field intensity can be written
H = φ̂H(ρ). Substituting this into Equation 7.10, we
obtain
Z 2π h
i φ̂H(ρ) · φ̂ ρ dφ
Iencl =
φ=0
= ρH(ρ)
Z 2π
dφ
φ=0
= 2πρH(ρ)
(7.9)
For the choice of C made above, Equation 7.7
becomes
Z 2π
H · φ̂ ρ dφ = Iencl
(7.10)
Therefore, H(ρ) = Iencl /2πρ. Reassociating the
known direction, we obtain:
Iencl
H = φ̂
(7.12)
2πρ
Therefore, the field outside of the wire is:
φ=0
Note that we have chosen to integrate in the +φ
direction. Therefore, the right-hand rule specifies that
positive Iencl corresponds to current flowing in the
+z direction, which is consistent with the direction
indicated in Figure 7.3. (Here’s an excellent exercise
to test your understanding. Change the direction of
the path of integration and confirm that you get the
same result obtained at the end of this section.
Changing the direction of integration should not
change the magnetic field associated with the
current!)
The simplest way to solve for H from Equation 7.10
is to use a symmetry argument, which proceeds as
follows:
• Since the distribution of current is uniform and
infinite in the z-dimension, H can’t depend on z,
and so H · ẑ must be zero everywhere.
• The problem is identical after any amount of
rotation in φ; therefore, the magnitude of H
cannot depend on φ. This is a form of radial
symmetry. Since we determined above that H
can’t depend on z either, it must be that the
magnitude of H can depend only on ρ.
(7.11)
H = φ̂
I
for ρ ≥ a
2πρ
(7.13)
whereas the field inside the wire is:
H = φ̂
Iρ
for ρ < a
2πa2
(7.14)
(By the way, this is a good time for a units check.)
Note that as ρ increases from zero to a (i.e., inside the
wire), the magnetic field is proportional to ρ and
therefore increases. However, as ρ continues to
increase beyond a (i.e., outside the wire), the
magnetic field is proportional to ρ−1 and therefore
decreases.
If desired, the associated magnetic flux density can be
obtained using B = µH.
Summarizing:
The magnetic field due to current in an infinite
straight wire is given by Equations 7.13 (outside
the wire) and 7.14 (inside the wire). The magnetic field is +φ̂-directed for current flowing in
the +z direction, so the magnetic field lines form
concentric circles perpendicular to and centered
on the wire.
CHAPTER 7. MAGNETOSTATICS
152
7.6 Magnetic Field Inside a
Straight Coil
I
[m0120]
H
c Jfmelero CC BY SA 4.0 (modified)
Figure 7.4: Right-hand rule for the relationship between the direction of current and the direction of the
magnetic field.
Finally, we point out another “right-hand rule” that
emerges from this solution, shown in Figure 7.4 and
summarized below:
The magnetic field due to current in an infinite
straight wire points in the direction of the curled
fingers of the right hand when the thumb of the
right hand is aligned in the direction of current
flow.
This simple rule turns out to be handy in quickly
determining the relationship between the directions of
the magnetic field and current flow in many other
problems, and so is well worth committing to
memory.
In this section, we use the magnetostatic integral form
of Ampere’s Circuital Law (ACL) (Section 7.4) to
determine the magnetic field inside a straight coil of
the type shown in Figure 7.5 in response to a steady
(i.e., DC) current. The result has a number of
applications, including the analysis and design of
inductors, solenoids (coils that are used as magnets,
typically as part of an actuator), and as a building
block and source of insight for more complex
problems.
The present problem is illustrated in Figure 7.6. The
coil is circular with radius a and length l, and consists
of N turns (“windings”) of wire wound with uniform
winding density. Since the coil forms a cylinder, the
problem is easiest to work in cylindrical coordinates
with the axis of the coil aligned along the z axis.
To begin, let’s take stock of what we already know
about the answer, which is actually quite a bit. The
magnetic field deep inside the coil is generally aligned
with axis of the coil as shown in Figure 7.7. This can
be explained using the result for the magnetic field
due to a straight line current (Section 7.5), in which
we found that the magnetic field follows a “right-hand
rule.” The magnetic field points in the direction of the
fingers of the right hand when the thumb of the right
hand is aligned in the direction of current flow. The
wire comprising the coil is obviously not straight, but
we can consider just one short segment of one turn
and then sum the results for all such segments. When
we consider this for a single turn of the coil, the
situation is as shown in Figure 7.8. Summing the
results for many loops, we see that the direction of the
magnetic field inside the coil must be generally in the
by Zureks (public domain)
Figure 7.5: A straight coil.
7.6. MAGNETIC FIELD INSIDE A STRAIGHT COIL
153
I
Figure 7.6: Determination of the magnetic field due
to DC current in a coil.
c Chetvorno CC0 1.0 (modified)
Figure 7.8: Magnetic field due to a single loop.
c Geek3 CC BY SA 3.0
Figure 7.7: Magnetic field lines inside a straight coil
with closely-spaced windings. (Dotted circles represent current flowing up/out from the page; crossed circles represent current flowing down/into the page.)
that the most general form for the magnetic field
intensity deep inside the coil is H ≈ ẑH(ρ). That is,
the direction of H is ±ẑ and the magnitude of H
depends, at most, on ρ. In fact, we will soon find with
the assistance of ACL that the magnitude of H
doesn’t depend on ρ either.
Here’s the relevant form of ACL:
I
H · dl = Iencl
(7.15)
C
+ẑ direction when the current I is applied as shown
in Figures 7.6 and 7.7. However, there is one caveat.
The windings must be sufficiently closely-spaced that
the magnetic field lines can only pass through the
openings at the end of the coil and do not take any
“shortcuts” between individual windings.
Figure 7.7 also indicates that the magnetic field lines
near the ends of the coil diverge from the axis of the
coil. This is understandable since magnetic field lines
form closed loops. The relatively complex structure
of the magnetic fields near the ends of the coil (the
“fringing field”) and outside of the coil make them
relatively difficult to analyze. Therefore, here we
shall restrict our attention to the magnetic field deep
inside the coil. This restriction turns out to be of little
consequence as long as l ≫ a.
Also, it is apparent from the radial symmetry of the
coil that the magnitude of the magnetic field cannot
depend on φ. Putting these findings together, we find
where Iencl is the current enclosed by the closed path
C. ACL works for any closed path that encloses the
current of interest. Also, for simplicity, we prefer a
path that lies on a constant-coordinate surface. The
selected path is shown in Figure 7.9. The benefits of
this particular path will soon become apparent.
However, note for now that this particular choice is
consistent with the right-hand rule relating the
direction of C to the direction of positive I. That is,
when I is positive, the current in the turns of the coil
pass through the surface bounded by C in the same
direction as the fingers of the right hand when the
thumb is aligned in the indicated direction of C.
Let’s define N to be the number of windings in the
coil. Then, the winding density of the coil is N/l
(turns/m). Let the path length in the z direction be l′ ,
as indicated in Figure 7.9. Then the enclosed current
is
N
(7.16)
Iencl = l′ I
l
That is, the number of turns per unit length times
CHAPTER 7. MAGNETOSTATICS
154
becomes:
C
D
N ′
lI=
l
[ẑH(ρ1 )] · (ẑdz)
Z z2
dz
= H(ρ1 )
ρ2
B
z1
z1
A
z
z1
Z z2
z2
ρ1
= H(ρ1 ) [z2 − z1 ]
Note z2 − z1 is simply l′ . Also, we have found that
the result is independent of ρ1 , as anticipated earlier.
Summarizing:
l'
H ≈ ẑ
cylindrical
form of coil
c K. Kikkeri CC BY SA 4.0
(7.18)
NI
inside coil
l
(7.19)
Let’s take a moment to consider the implications of
this remarkably simple result.
Figure 7.9: Selected path of integration.
length gives number of turns, and this quantity times
the current through the wire is the total amount of
current crossing the surface bounded by C.
For the choice of C made above, and taking our
approximation for the form of H as exact,
Equation 7.15 becomes
I
N
(7.17)
[ẑH(ρ)] · dl = l′ I
l
C
The integral consists of segments A, B, C, and D, as
shown in Figure 7.9. Let us consider the result for
each of these segments individually:
• The integral over segments B and D is zero
because dl = ρ̂dρ for these segments, and so
H · dl = 0 for these segments.
• It is also possible to make the contribution from
Segment C go to zero simply by letting
ρ2 → ∞. The argument is as follows. The
magnitude of H outside the coil must decrease
with distance from the coil, so for ρ sufficiently
large, H(ρ) becomes negligible. If that’s the
case, then the integral over Segment C also
becomes negligible.
• Note that it is dimensionally correct; that is,
current divided by length gives units of A/m,
which are the units of H.
• We have found that the magnetic field is simply
winding density (N/l) times current. To increase
the magnetic field, you can either use more turns
per unit length or increase the current.
• We have found that the magnetic field is uniform
inside the coil; that is, the magnetic field along
the axis is equal to the magnetic field close to the
cylinder wall formed by the coil. However, this
does not apply close to ends of the coil, since we
have neglected the fringing field.
These findings have useful applications in more
complicated and practical problems, so it is
worthwhile taking note of these now. Summarizing:
The magnetic field deep inside an ideal straight
coil (Equation 7.19) is uniform and proportional
to winding density and current.
Additional Reading:
• “Electromagnetic Coil” on Wikipedia.
With ρ2 → ∞, only Segment A contributes
significantly to the integral over C and Equation 7.17
• “Solenoid” on Wikipedia.
7.7. MAGNETIC FIELD OF A TOROIDAL COIL
155
y
7.7 Magnetic Field of a Toroidal
Coil
b
[m0049]
A toroid is a cylinder in which the ends are joined to
form a closed loop. An example of a toroidal coil is
shown in Figure 7.10. Toroidal coils are commonly
used to form inductors and transformers. The
principal advantage of toroidal coils over straight
coils in these applications is magnetic field
containment – as we shall see in this section, the
magnetic field outside of a toroidal coil can be made
negligibly small. This reduces concern about
interactions between this field and other fields and
structures in the vicinity.
In this section, we use the magnetostatic form of
Ampere’s Circuital Law (ACL) (Section 7.4) to
determine the magnetic field due to a steady (DC)
current flowing through a toroidal coil. The problem
is illustrated in Figure 7.11. The toroid is circular
with inner and outer radii a and b, respectively. The
coil consists of N windings (turns) of wire wound
with uniform winding density. This problem is easiest
to work in cylindrical coordinates with the toroid
centered on the origin in z = 0 plane.
To begin, let’s take stock of what we already know
about the answer, which is actually quite a bit. First, a
review of Section 7.6 (“Magnetic Field Inside a
Straight Coil”) is recommended. There it is shown
that the magnetic field deep inside a straight coil is
aligned with axis of the coil. This can be explained
I
x
a
H
Figure 7.11: Geometry of a toroidal coil.
using the result for the magnetic field due to a straight
line current (Section 7.5), in which we found that the
magnetic field follows a “right-hand rule” – The
magnetic field points in the direction of the curled
fingers of the right hand when the thumb of the right
hand is aligned in the direction of current flow. The
wire comprising the coil is obviously not straight, but
we can consider just one short segment of one turn
and then sum the results for all such segments. When
we do this, we see that the direction of the magnetic
field inside the coil must be in the +φ̂ direction when
the current I is applied as shown in Figure 7.11. Also,
because the problem is identical after any amount of
rotation around the z axis, the magnitude of the
magnetic field cannot depend on φ. Putting these
findings together, we find that the most general form
for the magnetic field intensity inside or outside the
coil is H = φ̂H(ρ, z).
Here’s the relevant form of ACL:
I
H · dl = Iencl
(7.20)
C
c Slick CC0 1.0
Figure 7.10: A toroidal coil used as a large-value inductor in the power supply of a wireless router.
where Iencl is the current enclosed by the closed path
C. ACL works for any closed path, but we need one
that encloses some current so as to obtain a
relationship between I and H. Also, for simplicity,
we prefer a path that lies on a constant-coordinate
surface. The selected path is a circle of radius ρ
CHAPTER 7. MAGNETOSTATICS
156
y
It is now clear that the result is independent of z0 .
Summarizing:
H = φ̂
b
NI
inside coil
2πρ
(7.24)
Let’s take a moment to consider the implications of
I
x this result.
ρ
• Note that it is dimensionally correct; that is,
current divided by the circumference of C (2πρ)
gives units of A/m, which are the units of H.
a
C
• We have found that the magnetic field is
proportional to winding density (i.e., number of
windings divided by circumference) times
current. To increase the magnetic field you can
either use more windings or increase the current.
Figure 7.12: Selected path of integration.
centered on the origin in the z = z0 plane, as shown
in Figure 7.12. We further require C to lie entirely
inside the coil, which ensures that the enclosed
current includes the current of all the windings as they
pass through the hole at the center of the toroid. We
choose the direction of integration to be in the +φ
direction, which is consistent with the indicated
direction of positive I. That is, when I is positive, the
current in the windings of the coil pass through the
surface bounded by C in the same direction as the
curled fingers of the right hand when the thumb is
aligned in the indicated direction of C.
In terms of the variables we have defined, the
enclosed current is simply
Iencl = N I
Equation 7.20 becomes
I h
i
φ̂H(ρ, z0 ) · dl = N I
(7.21)
(7.22)
C
Now evaluating the integral:
Z 2π h
i φ̂H(ρ, z0 ) · φ̂ ρ dφ
NI =
0
= ρH(ρ, z0 )
Z 2π
dφ
0
= 2πρH(ρ, z0 )
(7.23)
• Remarkably, we have found that the magnitude
of the magnetic field inside the coil depends only
on ρ; i.e., the distance from the central (here, z)
axis. It is independent of z.
Summarizing:
The magnetic field inside a toroidal coil (Equation 7.24) depends only on distance from the central axis and is proportional to winding density
and current.
Now let us consider what happens outside the coil.
For this, we consider any path of integration (C) that
lies completely outside the coil. Note that any such
path encloses no current and therefore Iencl = 0 for
any such path. In this case we have:
I
H · dl = 0
(7.25)
C
There are two ways this could be true. Either H could
be zero everywhere along the path, or H could be
non-zero along the path in such a way that the integral
windings out to be zero. The radial symmetry of the
problem rules out the second possibility – if H is
radially symmetric and C is radially symmetric, then
the sign of H · dl should not change over C.
Therefore:
7.8. MAGNETIC FIELD OF AN INFINITE CURRENT SHEET
The magnetic field everywhere outside an ideal
toroidal coil is zero.
Note the caveat signaled by the use of the adjective
“ideal.” In a practical toroidal coil, we expect there
will be some leakage of magnetic flux between the
windings. In practice, this leakage can be made
negligibly small by using a sufficiently high winding
density and winding the wire on material on a toroidal
form (a “core”) having sufficiently large permeability.
The use of a high-permeability core, as shown in
Figure 7.10, will dramatically improve the already
pretty-good containment. In fact, the use of such a
core allows the spacing between windings to become
quite large before leakage becomes significant.
One final observation about toroidal coils is that at no
point in the derivation of the magnetic field did we
need to consider the cross-sectional shape of the coil;
we merely needed to know whether C was inside or
outside the coil. Therefore:
The magnetic field inside an ideal toroidal coil
does not depend on the cross-sectional shape of
the coil.
Additional Reading:
• “Toroidal Inductors and Transformers” on
Wikipedia.
157
7.8 Magnetic Field of an Infinite
Current Sheet
[m0121]
We now consider the magnetic field due to an infinite
sheet of current, shown in Figure 7.13. The solution
to this problem is useful as a building block and
source of insight in more complex problems, as well
as being a useful approximation to some practical
problems involving current sheets of finite extent
including, for example, microstrip transmission line
and ground plane currents in printed circuit boards.
The current sheet in Figure 7.13 lies in the z = 0
plane and the current density is Js = x̂Js (units of
A/m); i.e., the current is uniformly distributed such
that the total current crossing any segment of width
∆y along the y direction is Js ∆y.
To begin, let’s take stock of what we already know
about the answer, which is actually quite a bit. For
example, imagine the current sheet as a continuum of
thin strips parallel to the x axis and very thin in the y
dimension. Each of these strips individually behaves
like a straight line current I = Js ∆y (units of A). The
magnetic field due to each of these strips is
determined by a “right-hand rule” – the magnetic field
points in the direction of the curled fingers of the right
hand when the thumb of the right hand is aligned in
the direction of current flow. (Section 7.5). It is
apparent from this much that H can have no ŷ
component, since the field of each individual strip has
z
C
Js
Lz
y
Ly
Figure 7.13: Analysis of the magnetic field due to an
infinite thin sheet of current.
CHAPTER 7. MAGNETOSTATICS
158
no ŷ component. When the magnetic field due to each
strip is added to that of all the other strips, the ẑ
component of the sum field must be zero due to
symmetry. It is also clear from symmetry
considerations that the magnitude of H cannot
depend on x or y. Summarizing, we have determined
that the most general form for H is ŷH(z), and
furthermore, the sign of H(z) must be positive for
z < 0 and negative for z > 0.
It’s possible to solve this problem by actually
summing over the continuum of thin current strips as
imagined above.1 However, it’s far easier to use
Ampere’s Circuital Law (ACL; Section 7.4). Here’s
the relevant form of ACL:
I
H · dl = Iencl
(7.26)
Therefore, only the horizontal sides contribute to the
integral and we have:
Lz
· (ŷdy)
ŷH −
2
−Ly /2
Z −Ly /2 Lz
ŷH +
+
· (ŷdy) = Js Ly (7.29)
2
+Ly /2
Z +Ly /2 Now evaluating the integrals:
Lz
Lz
Ly − H +
Ly = J s Ly
H −
2
2
(7.30)
Note that all factors of Ly cancel in the above
equation. Furthermore, H(−Lz /2) = −H(+Lz /2)
due to (1) symmetry between the upper and lower
half-spaces and (2) the change in sign between these
C
half-spaces, noted earlier. We use this to eliminate
where Iencl is the current enclosed by a closed path C.
H(+Lz /2) and solve for H(−Lz /2) as follows:
ACL works for any closed path, but we need one that
encloses some current so as to obtain a relationship
2H(−Lz /2) = Js
(7.31)
between Js and H. Also, for simplicity, we prefer a
path that lies on a constant-coordinate surface. A
yielding
Js
convenient path in this problem is a rectangle lying in
H(−Lz /2) = +
(7.32)
the x = 0 plane and centered on the origin, as shown
2
in Figure 7.13. We choose the direction of integration and therefore
to be counter-clockwise from the perspective shown
Js
in Figure 7.13, which is consistent with the indicated
H(+Lz /2) = −
(7.33)
2
direction of positive Js according to the applicable
right-hand rule from Stokes’ Theorem. That is, when Furthermore, note that H is independent of L ; for
z
Js is positive (current flowing in the +x̂ direction),
example, the result we just found indicates the same
the current passes through the surface bounded by C
value of H(+Lz /2) regardless of the value of Lz .
in the same direction as the curled fingers of the right Therefore, H is uniform throughout all space, except
hand when the thumb is aligned in the indicated
for the change of sign corresponding for the field
direction of C.
above vs. below the sheet. Summarizing:
Let us define Ly to be the width of the rectangular
path of integration in the y dimension and Lz to be
the width in the z dimension. In terms of the variables
we have defined, the enclosed current is simply
Iencl = Js Ly
Equation 7.26 becomes
I
[ŷH(z)] · dl = Js Ly
(7.27)
(7.28)
C
Note that H · dl = 0 for the vertical sides of the path,
since H is ŷ-directed and dl = ẑdz on those sides.
1 In fact, this is pretty good thing to try, if for no other reason than
to see how much simpler it is to use ACL instead.
H = ±ŷ
Js
for z ≶ 0
2
(7.34)
The magnetic field intensity due to an infinite
sheet of current (Equation 7.34) is spatially uniform except for a change of sign corresponding
for the field above vs. below the sheet.
7.9. AMPERE’S LAW (MAGNETOSTATICS): DIFFERENTIAL FORM
7.9 Ampere’s Law
(Magnetostatics): Differential
Form
159
possible for all possible surfaces in all applicable
scenarios is if the integrands are equal. Thus, we
obtain the desired expression:
∇×H=J
[m0118]
The integral form of Amperes’ Circuital Law (ACL;
Section 7.4) for magnetostatics relates the magnetic
field along a closed path to the total current flowing
through any surface bounded by that path. In
mathematical form:
I
H · dl = Iencl
(7.35)
C
where H is magnetic field intensity, C is the closed
curve, and Iencl is the total current flowing through
any surface bounded by C. In this section, we derive
the differential form of this equation. In some
applications, this differential equation, combined with
boundary conditions associated with discontinuities in
structure and materials, can be used to solve for the
magnetic field in arbitrarily complicated scenarios. A
more direct reason for seeking out this differential
equation is that we gain a little more insight into the
relationship between current and the magnetic field,
disclosed at the end of this section.
The equation we seek may be obtained using Stokes’
Theorem (Section 4.9), which in the present case may
be written:
I
Z
H · dl
(7.36)
(∇ × H) · ds =
C
S
where S is any surface bounded by C, and ds is the
differential surface area combined with the unit vector
in the direction determined by the right-hand rule
from Stokes’ Theorem. ACL tells us that the right
side of the above equation is simply Iencl . We may
express Iencl as the integral of the volume current
density J (units of A/m2 ; Section 6.2) as follows:
Z
Iencl =
J · ds
(7.37)
S
so we may rewrite Equation 7.36 as follows:
Z
Z
(∇ × H) · ds =
J · ds
S
(7.38)
S
The above relationship must hold regardless of the
specific location or shape of S. The only way this is
(7.39)
That is, the curl of the magnetic field intensity at a
point is equal to the volume current density at that
point. Recalling the properties of the curl operator
(Section 4.8) – in particular, that curl involves
derivatives with respect to direction – we conclude:
The differential form of Ampere’s Circuital Law
for magnetostatics (Equation 7.39) indicates that
the volume current density at any point in space
is proportional to the spatial rate of change of the
magnetic field and is perpendicular to the magnetic field at that point.
Additional Reading:
• “Ampere’s circuital law” on Wikipedia.
• “Boundary value problem” on Wikipedia.
CHAPTER 7. MAGNETOSTATICS
160
7.10 Boundary Conditions on the
Magnetic Flux Density (B)
[m0022]
In homogeneous media, electromagnetic quantities
vary smoothly and continuously. At an interface
between dissimilar media, however, it is possible for
electromagnetic quantities to be discontinuous.
Continuities and discontinuities in fields can be
described mathematically by boundary conditions and
used to constrain solutions for fields away from these
interfaces.
In this section, we derive the boundary condition on
the magnetic flux density B at a smooth interface
between two material regions, as shown in
Figure 7.14.2 The desired boundary condition may be
obtained from Gauss’ Law for Magnetic Fields
(GLM; Section 7.2):
I
S
B · ds = 0
(7.40)
where S is any closed surface. Let S take the form of
cylinder centered at a point on the interface, and for
which the flat ends are parallel to the surface and
perpendicular to n̂, as shown in Figure 7.14. Let the
radius of this cylinder be a, and let the length of the
2 It may be helpful to note the similarity (duality, in fact) between
this derivation and the derivation of the associated boundary condition on D presented in Section 5.18.
cylinder be 2h. From GLM, we have
I
Z
B · ds =
B · ds
S
top
Z
+
B · ds
side
Z
+
B · ds
=0
(7.41)
bottom
Now let us reduce h and a together while (1)
maintaining a constant ratio h/a ≪ 1 and (2) keeping
S centered on the interface. Because h ≪ a, the area
of the side can be made negligible relative to the area
of the top and bottom. Then, as h → 0, we are left
with
Z
Z
B · ds → 0
(7.42)
B · ds +
top
bottom
As the area of the top and bottom sides become
infinitesimal, the variation in B over these areas
becomes negligible. Now we have simply:
B1 · n̂∆A + B2 · (−n̂) ∆A → 0
(7.43)
where B1 and B2 are the magnetic flux densities at
the interface but in regions 1 and 2, respectively, and
∆A is the area of the top and bottom sides. Note that
the orientation of n̂ is important – we have assumed n̂
points into region 1, and we must now stick with this
choice. Thus, we obtain
n̂ · (B1 − B2 ) = 0
(7.44)
where, as noted above, n̂ points into region 1.
Summarizing:
a
region 1
n
h
h
region 2
c K. Kikkeri CC BY SA 4.0
Figure 7.14: Determination of the boundary condition
on B at the interface between material regions.
The normal (perpendicular) component of B
across the boundary between two material regions is continuous.
It is worth noting what this means for the magnetic
field intensity H. Since B = µH, it must be that
The normal (perpendicular) component of H
across the boundary between two material regions is discontinuous if the permeabilities are
unequal.
7.11. BOUNDARY CONDITIONS ON THE MAGNETIC FIELD INTENSITY (H)
7.11 Boundary Conditions on the
Magnetic Field Intensity (H)
[m0023]
In homogeneous media, electromagnetic quantities
vary smoothly and continuously. At a boundary
between dissimilar media, however, it is possible for
electromagnetic quantities to be discontinuous.
Continuities and discontinuities in fields can be
described mathematically by boundary conditions and
used to constrain solutions for fields away from these
boundaries. In this section, we derive boundary
conditions on the magnetic field intensity H.
To begin, consider a region consisting of only two
media that meet at a smooth boundary as shown in
Figure 7.15. The desired boundary condition can be
obtained directly from Ampere’s Circuital Law (ACL;
Section 7.4):
I
C
H · dl = Iencl
(7.45)
where C is any closed path and Iencl is the current
that flows through the surface bounded by that path in
the direction specified by the “right-hand rule” of
Stokes’ theorem. Let C take the form of a rectangle
centered on a point on the boundary as shown in
Figure 7.15, perpendicular to the direction of current
flow at that location. Let the sides A, B, C, and D be
perpendicular and parallel to the boundary. Let the
length of the parallel sides be l, and let the length of
n1
o
regi
A
n
t
D
w
Figure 7.15: Determining the boundary condition on
H at the smooth boundary between two material regions.
161
the perpendicular sides be w. Now we apply ACL.
We must integrate in a counter-clockwise direction in
order to be consistent with the indicated reference
direction for Js . Thus:
I
Z
H · dl =
H · dl
A
Z
+
H · dl
ZB
+
H · dl
ZC
+
H · dl
= Iencl
(7.46)
D
Now we let w and l become vanishingly small while
(1) maintaining the ratio l/w and (2) keeping C
centered on the boundary. In this process, the
contributions from the B and D segments become
equal in magnitude but opposite in sign; i.e.,
Z
Z
H · dl +
H · dl → 0
(7.47)
B
This leaves
Z
A
H · dl +
D
Z
C
H · dl → Iencl
(7.48)
Let us define the unit vector t̂ (“tangent”) as shown in
Figure 7.15. Now we have simply:
H1 · −t̂∆l + H2 · +t̂∆l = Iencl
(7.49)
where H1 and H2 are the fields evaluated on the two
sides of the boundary, and ∆l → 0 is the length of
sides A and C. As always, Iencl (units of A) may be
interpreted as the flux of the current density Js (units
of A/m) flowing past a line on the surface having
length ∆l (units of m) perpendicular to t̂ × n̂, where
n̂ is the normal to the surface, pointing into Region 1.
Stated mathematically:
Iencl → Js · ∆l t̂ × n̂
(7.50)
Before proceeding, note this is true regardless of the
particular direction we selected for t̂; it is only
necessary that t̂ be tangent to the boundary. Thus,
t̂ × n̂ need not necessarily be in the same direction as
Js . Now Equation 7.49 can be written:
H2 · t̂∆l − H1 · t̂∆l = Js · t̂ × n̂ ∆l
(7.51)
CHAPTER 7. MAGNETOSTATICS
162
Eliminating the common factor of ∆l and arranging
terms on the left:
(7.52)
(H2 − H1 ) · t̂ = Js · t̂ × n̂
The right side may be transformed using a vector
identity (Equation B.18 of Section B.3) to obtain:
(H2 − H1 ) · t̂ = t̂ · (n̂ × Js )
(7.53)
Equation 7.53 is the boundary condition we seek. We
have found that the component of H2 − H1 (the
difference between the magnetic field intensities at
the boundary) in any direction tangent to the
boundary is equal to the component of the current
density flowing in the perpendicular direction n̂ × Js .
Said differently:
A discontinuity in the tangential component of
the magnetic field intensity at the boundary must
be supported by surface current flowing in a direction perpendicular to this component of the
field.
An important consequence is that:
If there is no surface current, then the tangential
component of the magnetic field intensity is continuous across the boundary.
It is possible to obtain a mathematical form of the
boundary condition that is more concise and often
more useful than Equation 7.53. This form may be
obtained as follows. First, we note that the dot
product with respect to t̂ on both sides of
Equation 7.53 means simply “any component that is
tangent to the boundary.” We need merely to make
sure we are comparing the same tangential
component on each side of the equation. For example
n̂ × (H2 − H1 ) is tangential to the boundary, since n̂
is perpendicular to the boundary and therefore any
cross product involving n̂ will be perpendicular to n̂.
The corresponding component of the current density
is n̂ × (n̂ × Js ), so Equation 7.53 may be
equivalently written as follows:
n̂ × (H2 − H1 ) = n̂ × n̂ × Js
(7.54)
Applying a vector identity (Equation B.19 of
Section B.3) to the right side of Equation 7.54 we
obtain:
n̂ × n̂ × Js = n̂ (n̂ · Js ) − Js (n̂ · n̂)
= n̂ (0) − Js (1)
= −Js
(7.55)
Therefore:
n̂ × (H2 − H1 ) = −Js
(7.56)
The minus sign on the right can be eliminated by
swapping H2 and H1 on the left, yielding
n̂ × (H1 − H2 ) = Js
(7.57)
This is the form in which the boundary condition is
most commonly expressed.
It is worth noting what this means for the magnetic
field intensity B. Since B = µH:
In the absence of surface current, the tangential
component of B across the boundary between
two material regions is discontinuous if the permeabilities are unequal.
7.12. INDUCTANCE
7.12 Inductance
[m0123]
Current creates a magnetic field, which subsequently
exerts force on other current-bearing structures. For
example, the current in each winding of a coil exerts a
force on every other winding of the coil. If the
windings are fixed in place, then this force is unable
to do work (i.e., move the windings), so instead the
coil stores potential energy. This potential energy can
be released by turning off the external source. When
this happens, charge continues to flow, but is now
propelled by the magnetic force. The potential energy
that was stored in the coil is converted to kinetic
energy and subsequently used to redistribute the
charge until no current flows. At this point, the
inductor has expended its stored energy. To restore
energy, the external source must be turned back on,
restoring the flow of charge and thereby restoring the
magnetic field.
Now recall that the magnetic field is essentially
defined in terms of the force associated with this
potential energy; i.e., F = qv × B where q is the
charge of a particle comprising the current, v is the
velocity of the particle, and B is magnetic flux density
(Section 2.5). So, rather than thinking of the potential
energy of the system as being associated with the
magnetic force applied to current, it is equally valid to
think of the potential energy as being stored in the
magnetic field associated with the current distribution.
The energy stored in the magnetic field depends on
the geometry of the current-bearing structure and the
permeability of the intervening material because the
magnetic field depends on these parameters.
The relationship between current applied to a
structure and the energy stored in the associated
magnetic field is what we mean by inductance. We
may fairly summarize this insight as follows:
Inductance is the ability of a structure to store energy in a magnetic field.
The inductance of a structure depends on the geometry of its current-bearing structures and the
permeability of the intervening medium.
163
Note that inductance does not depend on current,
which we view as either a stimulus or response from
this point of view. The corresponding response or
stimulus, respectively, is the magnetic flux associated
with this current. This leads to the following
definition:
Φ
(single linkage)
(7.58)
L=
I
where Φ (units of Wb) is magnetic flux, I (units of
A) is the current responsible for this flux, and L (units
of H) is the associated inductance. (The “single
linkage” caveat will be explained below.) In other
words, a device with high inductance generates a
large magnetic flux in response to a given current, and
therefore stores more energy for a given current than a
device with lower inductance.
To use Equation 7.58 we must carefully define what
we mean by “magnetic flux” in this case. Generally,
magnetic flux is magnetic flux density (again, B, units
of Wb/m2 ) integrated over a specified surface S, so
Z
Φ=
B · ds
(7.59)
S
where ds is the differential surface area vector, with
direction normal to S. However, this leaves
unanswered the following questions: Which S, and
which of the two possible normal directions of ds?
For a meaningful answer, S must uniquely associate
the magnetic flux to the associated current. Such an
association exists if we require the current to form a
closed loop. This is shown in Figure 7.16. Here C is
the closed loop along which the current flows, S is a
surface bounded by C, and the direction of ds is
defined according to the right-hand rule of Stokes’
Theorem (Section 4.9). Note that C can be a closed
loop of any shape; i.e., not just circular, and not
restricted to lying in a plane. Further note that S used
in the calculation of Φ can be any surface bounded by
C. This is because magnetic field lines form closed
loops such that any one magnetic field line intersects
any open surface bounded by C exactly once. Such an
intersection is sometimes called a “linkage.” So there
we have it – we require the current I to form a closed
loop, we measure the magnetic flux through this loop
using the sign convention of the right-hand rule, and
the ratio is the inductance.
Many structures consist of multiple such loops – the
coil is of course one of these. In a coil, each winding
CHAPTER 7. MAGNETOSTATICS
164
B
ds
I
An inductor is a device that is designed to exhibit a
specified inductance. We can now make the
connection to the concept of the inductor as it appears
in elementary circuit theory. First, we rewrite
Equation 7.60 as follows:
I=
NΦ
L
(7.61)
Taking the derivative of both sides of this equation
with respect to time, we obtain:
c K. Kikkeri CC BY SA 4.0
Figure 7.16: Association between a closed loop of
current and the associated magnetic flux.
carries the same current, and the magnetic fields of the
windings add to create a magnetic field, which grows
in proportion to the winding density (Section 7.6).
The magnetic flux density inside a coil is proportional
to the number of windings, N , so the flux Φ in
Equation 7.58 should properly be indicated as N Φ.
Another way to look at this is that we are counting the
number of times the same current is able to generate a
unique set of magnetic field lines that intersect S.
Summarizing, our complete definition for inductance
is
NΦ
(identical linkages)
(7.60)
L=
I
An engineering definition of inductance is Equation 7.60, with the magnetic flux defined to be
that associated with a single closed loop of current with sign convention as indicated in Figure 7.16, and N defined to be the number of times
the same current I is able to create that flux.
What happens if the loops have different shapes? For
example, what if the coil is not a cylinder, but rather
cone-shaped? (Yes, there is such a thing – see
“Additional Reading” at the end of this section.) In
this case, one needs a better way to determine the
factor N Φ since the flux associated with each loop of
current will be different. However, this is beyond the
scope of this section.
N d
d
I=
Φ
dt
L dt
(7.62)
Now we need to reach beyond the realm of
magnetostatics for just a moment. Section 8.3
(“Faraday’s Law”) shows that the change in Φ
associated with a change in current results in the
creation of an electrical potential equal to −N dΦ/dt
realized over the loop C. In other words, the terminal
voltage V is +N dΦ/dt, with the change of sign
intended to keep the result consistent with the sign
convention relating current and voltage in passive
devices. Therefore, dΦ/dt in Equation 7.62 is equal
to V /N . Making the substitution we find:
V =L
d
I
dt
(7.63)
This is the expected relationship from elementary
circuit theory.
Another circuit theory concept related to inductance
is mutual inductance. Whereas inductance relates
changes in current to instantaneous voltage in the
same device (Equation 7.63), mutual inductance
relates changes in current in one device to
instantaneous voltage in a different device. This can
occur when the two devices are coupled (“linked”) by
the same magnetic field. For example, transformers
(Section 8.5) typically consist of separate coils that
are linked by the same magnetic field lines. The
voltage across one coil may be computed as the
time-derivative of current on the other coil times the
mutual inductance.
Let us conclude this section by taking a moment to
dispel a common misconception about inductance.
The misconception pertains to the following question.
If the current does not form a closed loop, what is the
inductance? For example, engineers sometimes refer
7.13. INDUCTANCE OF A STRAIGHT COIL
to the inductance of a pin or lead of an electronic
component. A pin or lead is not a closed loop, so the
formal definition of inductance given above – ratio of
magnetic flux to current – does not apply. The
broader definition of inductance – the ability to store
energy in a magnetic field – does apply, but this is not
what is meant by “pin inductance” or “lead
inductance.” What is actually meant is the imaginary
part of the impedance of the pin or lead – i.e., the
reactance – expressed as an equivalent inductance. In
other words, the reactance of an inductive device is
positive, so any device that also exhibits a positive
reactance can be viewed from a circuit theory
perspective as an equivalent inductance. This is not
referring to the storage of energy in a magnetic field;
it merely means that the device can be modeled as an
inductor in a circuit diagram. In the case of “pin
inductance,” the culprit is not actually inductance, but
rather skin effect (see “Additional References” at the
end of this section). Summarizing:
165
7.13 Inductance of a Straight
Coil
[m0124]
In this section, we determine the inductance of a
straight coil, as shown in Figure 7.17. The coil is
circular with radius a and length l and consists of N
windings of wire wound with uniform winding
density. Also, we assume the winding density N/l is
large enough that magnetic field lines cannot enter or
exit between windings but rather must traverse the
entire length of the coil. Since the coil forms a
cylinder, the problem is easiest to work in cylindrical
coordinates with the axis of the coil aligned along the
z axis.
Inductance L in this case is given by (Section 7.12)
L=
Inductance implies positive reactance, but positive reactance does not imply the physical mechanism of inductance.
Additional Reading:
NΦ
I
(7.64)
where I is current and Φ is the magnetic flux
associated with one winding of the coil. Magnetic
flux in this case is given by
Z
Φ=
B · ds
(7.65)
S
• “Inductance” on Wikipedia.
• “Inductor” on Wikipedia.
• T.A. Winslow,
“Conical Inductors for Broadband Applications,”
IEEE Microwave Mag., Vol. 6, No. 1, Mar 2005,
pp. 68–72.
where B is the magnetic flux density (units of T =
Wb/m2 ), S is the surface bounded by a single current
loop, and ds points in the direction determined by the
right hand rule with respect to the direction of
positive current flow.
First, let’s determine the magnetic field. The magnetic
• “Skin Effect” on Wikipedia.
Figure 7.17:
straight coil.
Determination of the inductance of a
CHAPTER 7. MAGNETOSTATICS
166
flux density deep inside the coil is (Section 7.6):
B ≈ ẑ
µN I
l
(7.66)
Is it reasonable to use this approximation here? Since
inductance pertains to energy storage, the question is
really what fraction of the energy is stored in a field
that is well-described by this approximation, as
opposed to energy stored in the “fringing field” close
to the ends of the coil. If we make l sufficiently large
relative to a, then presumably energy storage in the
fringing field will be negligible in comparison. Since
the alternative leads to a much more complicated
problem, we shall assume that Equation 7.66 is valid
for the interior of the coil.
Next, we determine Φ. In this case, a natural choice
for S is the interior cross-section of the coil in a plane
perpendicular to the axis. The direction of ds must be
+ẑ since this is the direction in which the fingers of
the right hand point when the current flows in the
direction indicated in Figure 7.17. Thus, we have
Z µN I
ẑ
l
S
Z
µN I
=
ds
l
S
µN I
=
A
l
Φ≈
· (ẑds)
(7.67)
where A is the cross-sectional area of the coil.
Finally from Equation 7.64 we obtain
L≈
The inductance of a long straight coil is given approximately by Equation 7.68.
µN 2 A
(l ≫ a)
l
(7.68)
Note that this is dimensionally correct; that is,
permeability (units of H/m) times area (units of m2 )
divided by length (units of m) gives units of H, as
expected. Also, it is worth noting that inductance is
proportional to permeability and cross-sectional area,
and inversely proportional to length. Interestingly the
inductance is proportional to N 2 as opposed to N ;
this is because field strength increases with N , and
independently there are N flux linkages. Finally, we
note that the inductance does not depend on the shape
of the coil cross-section, but only on the area of the
cross-section. Summarizing:
Again, this result is approximate because it neglects
the non-uniform fringing field near the ends of the
coil and the possibility that magnetic field lines
escape between windings due to inadequate winding
density. Nevertheless, this result facilitates useful
engineering analysis and design.
Additional Reading:
• “Inductance” on Wikipedia.
7.14. INDUCTANCE OF A COAXIAL STRUCTURE
7.14 Inductance of a Coaxial
Structure
[m0125]
Let us now determine the inductance of coaxial
structure, shown in Figure 7.18. The inductance of
this structure is of interest for a number of reasons –
in particular, for determining the characteristic
impedance of coaxial transmission line, as addressed
in Section 3.10.
For our present purpose, we may model the structure
as shown in Figure 7.18. This model consists of two
concentric perfectly-conducting cylinders of radii a
and b, separated by a homogeneous material having
permeability µ. To facilitate analysis, let us place the
+z axis along the common axis of the concentric
cylinders, so that the cylinders may be described as
the surfaces ρ = a and ρ = b.
Below we shall find the inductance by assuming a
current I on the inner conductor and integrating over
the resulting magnetic field to obtain the magnetic
flux Φ between the conductors. Then, inductance can
be determined as the ratio of the response flux to the
source current.
Before we get started, note the derivation we are
about to do is similar to the derivation of the
capacitance of a coaxial structure, addressed in
Section 5.24. The reader may benefit from a review of
that section before attempting this derivation.
ρ
μ
H
b
z
a
I
z=0
z=l
Figure 7.18: Determining the inductance of coaxial
line.
167
The first step is to find the magnetic field inside the
structure. This is relatively simple if we may neglect
fringing fields, since then the internal field may be
assumed to be constant with respect to z. This
analysis will also apply to the case where the length l
pertains to one short section of a much longer
structure; in this case we will obtain the inductance
per length as opposed to the total inductance for the
structure. Note that the latter is exactly what we need
for the transmission line lumped-element equivalent
circuit model (Section 3.4).
To determine the inductance, we invoke the definition
(Section 7.12):
Φ
L,
(7.69)
I
A current I flowing in the +z direction on the inner
conductor gives rise to a magnetic field inside the
coaxial structure. The magnetic field intensity for this
scenario was determined in Section 7.5 where we
found
I
, a≤ρ≤b
(7.70)
H = φ̂
2πρ
The reader should note that in that section we were
considering merely a line of current; not a coaxial
structure. So, on what basis do we claim the field for
inside the coaxial structure is the same? This is a
consequence of Ampere’s Law (Section 7.4):
I
H · dl = Iencl
(7.71)
C
If in this new problem we specify the same circular
path C with radius greater than a and less than b, then
the enclosed current is simply I. The presence of the
outer conductor does not change the radial symmetry
of the problem, and nothing else remains that can
change the outcome. This is worth noting for future
reference:
The magnetic field inside a coaxial structure
comprised of concentric conductors bearing current I is identical to the magnetic field of the line
current I in free space.
We’re going to need magnetic flux density (B) as
opposed to H in order to get the magnetic flux. This
is simple since they are related by the permeability of
the medium; i.e., B = µH. Thus:
B = φ̂
µI
, a≤ρ≤b
2πρ
(7.72)
CHAPTER 7. MAGNETOSTATICS
168
Next, we get Φ by integrating over the magnetic flux
density
Z
Φ=
S
B · ds
(7.73)
where S is any open surface through which all
magnetic field lines within the structure must pass.
Since this can be any such surface, we may as well
choose the simplest one. The simplest such surface is
a plane of constant φ, since such a plane is a
constant-coordinate surface and perpendicular to the
magnetic field lines. This surface is shown as the
shaded area in Figure 7.18. Using this surface we
find:
Z b Z l µI
· φ̂ dρ dz
φ̂
Φ=
2πρ
ρ=a z=0
! Z
!
Z l
b
µI
dρ
=
dz
2π
ρ=a ρ
z=0
b
µIl
(7.74)
ln
=
2π
a
Wrapping up:
L,
Φ
(µIl/2π) ln (b/a)
=
I
I
(7.75)
Note that factors of I in the numerator and
denominator cancel out, leaving:
µl
ln
L=
2π
b
a
(7.76)
Note that this is dimensionally correct, having units of
H. Also note that this is expression depends only on
materials (through µ) and geometry (through l, a, and
b). Notably, it does not depend on current, which
would imply non-linear behavior.
To make the connection back to lumped-element
transmission line model parameters (Sections 3.4 and
3.10), we simply divide by l to get the per-unit length
parameter:
b
µ
(7.77)
ln
L′ =
2π
a
which has the expected units of H/m.
Example 7.1. Inductance of RG-59 coaxial
cable.
RG-59 coaxial cable consists of an inner
conductor having radius 0.292 mm, an outer
conductor having radius 1.855 mm, and
polyethylene (a non-magnetic dielectric) spacing
material. Estimate the inductance per length of
RG-59.
Solution. From the problem statement,
a = 0.292 mm, b = 1.855 mm, and µ ∼
= µ0
since the spacing material is non-magnetic.
Using Equation 7.77, we find L′ ∼
= 370 nH/m.
7.15. MAGNETIC ENERGY
169
7.15 Magnetic Energy
the present value of the magnetic energy is:
Wm =
[m0127]
Z t0 +t
p(τ )dτ
(7.80)
t0
Consider a structure exhibiting inductance; i.e., one
that is able to store energy in a magnetic field in
response to an applied current. This structure could
be a coil, or it could be one of a variety of inductive
structures that are not explicitly intended to be an
inductor; for example, a coaxial transmission line.
When current is applied, the current-bearing elements
of the structure exert forces on each other. Since these
elements are not normally free to move, we may
interpret this force as potential energy stored in the
magnetic field associated with the current
(Section 7.12).
We now want to know how much energy is stored in
this field. The answer to this question has relevance in
several engineering applications. One issue is that any
system that includes inductance is using some
fraction of the energy delivered by the power supply
to energize this inductance. In many electronic
systems – in power systems in particular – inductors
are periodically energized and de-energized at a
regular rate. Since power is energy per unit time, this
consumes power. Therefore, energy storage in
inductors contributes to the power consumption of
electrical systems.
The stored energy is most easily determined using
circuit theory concepts. First, we note that the
electrical potential difference v(t) (units of V) across
an inductor is related to the current i(t) (units of A)
through the inductor as follows (Section 7.12):
v(t) = L
d
i(t)
dt
(7.78)
where L (units of H) is the inductance. The
instantaneous power associated with the device is
Now evaluating this integral using the relationships
established above:
Z t+t0
v(τ )i(τ )dτ
Wm =
t0
Z t+t0 d
L i(τ ) i(τ )dτ
=
dτ
t0
Z t+t0 d
=L
i(τ ) i(τ )dτ
dτ
t0
Changing the variable of integration from τ (and dτ )
to i (and di) we have
Wm = L
Z t+t0
t0
=L
Z I
(7.79)
Energy (units of J) is power (units of J/s) integrated
over time. Let Wm be the energy stored in the
inductor. At some time t0 in the past, i(t0 ) = 0 and
Wm = 0. As current is applied, Wm increases
monotonically. At the present time t, i(t) = I. Thus,
di
i dτ
dτ
i di
(7.82)
0
Evaluating the integral we obtain the desired
expression
1
Wm = LI 2
2
(7.83)
The energy stored in an inductor in response to
a steady current I is Equation 7.83. This energy
increases in proportion to inductance and in proportion to the square of current.
The long straight coil (Section 7.13) is representative
of a large number of practical applications, so it is
useful to interpret the above findings in terms of this
structure in particular. For this structure we found
L=
p(t) = v(t)i(t)
(7.81)
µN 2 A
l
(7.84)
where µ is the permeability, N is the number of
windings, A is cross-sectional area, and l is length.
The magnetic field intensity inside this structure is
related to I by (Section 7.6):
H=
NI
l
(7.85)
CHAPTER 7. MAGNETOSTATICS
170
Substituting these expressions into Equation 7.83, we
obtain
2
1 µN 2 A Hl
Wm =
2
l
N
1
(7.86)
= µH 2 Al
2
Recall that the magnetic field inside a long coil is
approximately uniform. Therefore, the density of
energy stored inside the coil is approximately
uniform. Noting that the product Al is the volume
inside the coil, we find that this energy density is
Wm /Al; thus:
1
(7.87)
wm = µH 2
2
which has the expected units of energy per unit
volume (J/m3 ).
The above expression provides an alternative method
to compute the total magnetostatic energy in any
structure. Within a mathematical volume V, the total
magnetostatic energy is simply the integral of the
energy density over V; i.e.,
Z
Wm =
wm dv
(7.88)
V
This works even if the magnetic field and the
permeability vary with position. Substituting
Equation 7.87 we obtain:
Wm =
1
2
Z
µH 2 dv
(7.89)
V
Summarizing:
The energy stored by the magnetic field present
within any defined volume is given by Equation 7.89.
It’s worth noting that this energy increases with the
permeability of the medium, which makes sense since
inductance is proportional to permeability.
Finally, we reiterate that although we arrived at this
result using the example of the long straight coil,
Equations 7.87 and 7.89 are completely general.
7.16 Magnetic Materials
[m0058]
As noted in Section 2.5, magnetic fields arise in the
presence of moving charge (i.e., current) and in the
presence of certain materials. In this section, we
address these “magnetic materials.”
A magnetic material may be defined as a substance
that exhibits permeability µ (Section 2.6) that is
significantly different from the permeability of free
space µ0 . Since the magnetic flux density B is related
to the magnetic field intensity H via B = µH,
magnetic materials may exhibit magnetic flux density
in response to a given magnetic field intensity that is
significantly greater than that of other materials.
Magnetic materials are also said to be
“magnetizable,” meaning that the application of a
magnetic field causes the material itself to become a
source of the magnetic field.
Magnetic media are typically metals, semiconductors,
or heterogeneous media containing such materials.
An example is ferrite, which consists of iron particles
suspended in a ceramic. Magnetic media are
commonly classified according to the physical
mechanism responsible for their magnetizability.
These mechanisms include paramagnetism,
diamagnetism, and ferromagnetism. All three of these
mechanisms involve quantum mechanical processes
operating at the atomic and subatomic level, and are
not well-explained by classical physics. These
processes are beyond the scope of this book (but
information is available via “Additional References”
at the end of this section). However, it is possible to
identify some readily-observable differences between
these categories of magnetic media.
Paramagnetic and diamagnetic materials exhibit
permeability that is only very slightly different than
µ0 and typically by much less than 0.01%. These
materials exhibit very weak and temporary
magnetization. The principal distinction between
paramagnetic and diamagnetic media is in the
persistence and orientation of induced magnetic
fields. Paramagnetic materials – including aluminum,
magnesium, and platinum – exhibit a very weak
persistent magnetic field, and the magnetic field
7.16. MAGNETIC MATERIALS
induced in the material is aligned in the same
direction as the impressed (external) magnetic field.
Diamagnetic materials – including copper, gold, and
silicon – do not exhibit a persistent magnetic field,
and the magnetic field induced in the material is
(counter to intuition!) aligned in the opposite
direction as the impressed magnetic field. The
magnetization of paramagnetic and diamagnetic
media is typically so weak that it is not often a
consideration in engineering analysis and design.
171
B
saturation
H
Paramagnetic and diamagnetic media exhibit permeability only very slightly different than that of
free space, with little or no magnetization.
c Ndthe (modified) CC BY SA 3.0
Ferromagnetic materials, on the other hand, exhibit
permeability that can be many orders of magnitude
greater than µ0 . (See Appendix A.2 for some example
values.) These materials can be readily and
indefinitely magnetized, thus, permanent magnets are
typically comprised of ferromagnetic materials.
Commonly-encountered ferromagnetic materials
include iron, nickel, and cobalt.
Ferromagnetic materials are significantly non-linear
(see definition in Section 2.8), exhibiting saturation
and hysteresis. This is illustrated in Figure 7.19. In
this plot, the origin represents a ferromagnetic
material that is unmagnetized and in a region free of
an external magnetic field. The external magnetic
field is quantified in terms of H, plotted along the
horizontal axis. As the external field is increased, so
to is B in the material, according to the relationship
B = µH. Right away we see the material is
non-linear, since the slope of the curve – and hence µ
– is not constant.
Once the external magnetizing field H exceeds a
certain value, the response field B no longer
significantly increases. This is saturation. Once
saturated, further increases in the external field result
do not significantly increase the magnetization of the
material, so there is no significant increase in B.
From this state of saturation, let us now reduce the
external field. We find that the rate of decrease in B
with respect to H is significantly less than the rate
that B originally increased with respect to H. In fact,
B is still greater than zero even when H has been
reduced to zero. At this point, the magnetization of
Figure 7.19: Non-linearity in a ferromagnetic material
manifesting as saturation and hysteresis.
the material is obvious, and a device comprised of this
material could be used as a magnet.
If we now apply an external field in the reverse
direction, we find that we are eventually able to zero
and then redirect the response field. As we continue
to decrease H (that is, increase the magnitude in the
reverse direction), we once again reach saturation.
The same behavior is observed when we once again
increase H. The material is eventually demagnetized,
remagnetized in the opposite direction and then
saturated in that direction. At this point, it is apparent
that a return to the start condition (H = B = 0; i.e.,
demagnetized when there is no external field) is not
possible.
Hysteresis is the name that we apply to this particular
form of non-linear behavior. Hysteresis has important
implications in engineering applications. First, as
identified above, it is an important consideration in
the analysis and design of magnets. In applications
where a ferromagnetic material is being used because
high permeability is desired – e.g., in inductors
(Section 7.12) and transformers (Section 8.5) –
hysteresis complicates the design and imposes limits
on the performance of the device.
Hysteresis may also be exploited as a form of
memory. This is apparent from Figure 7.19. If B > 0,
CHAPTER 7. MAGNETOSTATICS
172
then recent values of H must have been relatively
large and positive. Similarly, If B < 0, then recent
values of H must have been relatively large and
negative. Furthermore, the most recent sign of H can
be inferred even if the present value of H is zero. In
this sense, the material “remembers” the past history
of its magnetization and thereby exhibits memory.
This is the enabling principle for a number of digital
data storage devices, including hard drives (see
“Additional Reading” at the end of this section).
Summarizing:
Ferromagnetic media exhibit permeability µ that
is orders of magnitude greater than that of free
space and are readily magnetizable. These materials are also nonlinear in µ, which manifests as
saturation and hysteresis.
Additional Reading:
• Section A.2 (“Permeability of Some Common
Materials”)
• “Magnetism” on Wikipedia.
• “Ferrite (magnet)” on Wikipedia.
• “Paramagnetism” on Wikipedia.
• “Diamagnetism” on Wikipedia.
• “Ferromagnetism” on Wikipedia.
• “Magnetic hysteresis” on Wikipedia.
• “Magnetic storage” on Wikipedia.
• “Hard disk drive” on Wikipedia.
[m0150]
7.16. MAGNETIC MATERIALS
Image Credits
Fig. 7.1: c Youming / K. Kikkeri,
https://commons.wikimedia.org/wiki/File:M0018 fGLMBarMagnet (2).svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 7.2: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0019 fACL.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 7.3: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0119 fLSW.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 7.4: c Jfmelero, https://et.wikipedia.org/wiki/Fail:Manoderecha.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/), modified.
Fig. 7.5: Zureks, https://en.wikipedia.org/wiki/File:Solenoid-1.png,
public domain.
Fig. 7.7: c Geek3, https://commons.wikimedia.org/wiki/File:VFPt Solenoid correct.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Fig. 7.8: Chetvorno, https://en.wikipedia.org/wiki/File:Magnetic field of loop 3.svg,
public domain via CC0 1.0, modified.
Fig. 7.9: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0120 fPath.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 7.10: Slick, https://en.wikipedia.org/wiki/File:
3Com OfficeConnect ADSL Wireless 11g Firewall Router 2012-10-28-0869.jpg,
public domain via CC0 1.0 (https://creativecommons.org/publicdomain/zero/1.0/deed.en).
Fig. 7.14: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0022 fBoundary.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 7.16: c K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0123 fFluxCurrent.svg,
CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 7.19: c Ndthe, https://commons.wikimedia.org/wiki/File:HysteresisSVG.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/deed.en), modified by author.
173
Chapter 8
Time-Varying Fields
8.1 Comparison of Static and
Time-Varying
Electromagnetics
• “Maxwell’s Equations” on Wikipedia.
[m0013]
Students encountering time-varying electromagnetic
fields for the first time have usually been exposed to
electrostatics and magnetostatics already. These
disciplines exhibit many similarities as summarized in
Table 8.1. The principles of time-varying
electromagnetics presented in this table are all
formally introduced in other sections; the sole
purpose of this table is to point out the differences.
We can summarize the differences as follows:
Maxwell’s Equations in the general (timevarying) case include extra terms that do not appear in the equations describing electrostatics and
magnetostatics. These terms involve time derivatives of fields and describe coupling between
electric and magnetic fields.
The coupling between electric and magnetic fields in
the time-varying case has one profound consequence
in particular. It becomes possible for fields to
continue to exist even after their sources – i.e.,
charges and currents – are turned off. What kind of
field can continue to exist in the absence of a source?
Such a field is commonly called a wave. Examples of
waves include signals in transmission lines and
signals propagating away from an antenna.
Additional Reading:
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
8.2. ELECTROMAGNETIC INDUCTION
Electric & magnetic
fields are...
Maxwell’s Eqns.
(integral)
Maxwell’s Eqns.
(differential)
175
Electrostatics /
Magnetostatics
independent
Time-Varying
(Dynamic)
possibly coupled
H
HS D · ds = Qencl
HC E · dl = 0
HS B · ds = 0
H · ds = Iencl
C
∇ · D = ρv
∇×E=0
∇·B=0
∇×H=J
H
HS D · ds = Q∂encl
R
HC E · dl = − ∂t S B · ds
HS B · ds = 0
R ∂
H · dl = Iencl + S ∂t
D · ds
C
∇ · D = ρv
∂
∇ × E = − ∂t
B
∇·B=0
∂
∇ × H = J+ ∂t
D
Table 8.1: Comparison of principles governing static and time-varying electromagnetic fields. Differences in
the time-varying case relative to the static case are highlighted in blue.
8.2 Electromagnetic Induction
[m0129]
When an electrically-conducting structure is exposed
to a time-varying magnetic field, an electrical
potential difference is induced across the structure.
This phenomenon is known as electromagnetic
induction. A convenient introduction to
electromagnetic induction is provided by Lenz’s Law.
This section explains electromagnetic induction in the
context of Lenz’s Law and provides two examples.
Y. Qin (modified) CC BY 4.0
Figure 8.1: An experiment demonstrating electromagnetic induction and Lenz’s Law.
Let us begin with the example depicted in Figure 8.1,
involving a cylindrical coil. Attached to the terminals
of the coil is a resistor for which we may identify an
electric potential difference V and current I. In this
particular case, the sign conventions indicated for V
and I are arbitrary, but it is important to be consistent
once they are established.
two of these tests that current flows (i.e., |I| > 0), and
subsequently, there is an induced magnetic field Bind
due to this current. It is the direction of the current
and subsequently the direction of Bind inside the coil
that we wish to observe. The findings are summarized
below and in Table 8.2.
Now let us introduce a bar magnet as shown in
Figure 8.1. The magnet is centered along the axis of
coil, to the right of the coil, and with its north pole
facing toward the coil. The magnet is responsible for
the magnetic flux density Bimp . We refer to Bimp as
an impressed magnetic field because this field exists
independently of any response that may be induced
by interaction with the coil. Note that Bimp points to
the left inside the coil.
• When the magnet is motionless, we have the
unsurprising result that there is no current in the
coil. Therefore, no magnetic field is induced, and
the total magnetic field is simply equal to Bimp .
The experiment consists of three tests. We will find in
• When the magnet moves toward the coil, we
observe current that is positive with respect to
the reference direction indicated in Figure 8.1.
This current creates an induced magnetic field
Bind that points to the right, as predicted by
magnetostatic considerations from the right-hand
CHAPTER 8. TIME-VARYING FIELDS
176
rule. Since Bimp points to the left, it appears
that the induced current is opposing the increase
in the magnitude of the total magnetic field.
• When the magnet moves away from the coil, we
observe current that is negative with respect to
the reference direction indicated in Figure 8.1.
This current yields Bind that points to the left.
Since Bimp points to the left, it appears that the
induced current is opposing the decrease in the
magnitude of the total magnetic field.
The first conclusion one may draw from this
experiment is that changes in the magnetic field can
induce current. This was the claim made in the first
paragraph of this section and is a consequence of
Faraday’s Law, which is tackled in detail in
Section 8.3. The second conclusion – also associated
with Faraday’s Law – is the point of this section: The
induced magnetic field – that is, the one due to the
current induced in the coil – always opposes the
change in the impressed magnetic field. Generalizing:
Lenz’s Law states that the current that is induced
by a change in an impressed magnetic field creates an induced magnetic field that opposes (acts
to reduce the effect of) the change in the total
magnetic field.
When the magnet moves, three things happen: (1) A
current is induced, (2) A magnetic field is induced
(which adds to the impressed magnetic field), and (3)
the value of V becomes non-zero. Lenz’s Law does
not address which of these are responding directly to
the change in the impressed magnetic field, and which
of these are simply responding to changes in the other
quantities. Lenz’s Law may leave you with the
incorrect impression that it is I that is induced, and
that Bind and V are simply responding to this current.
In truth, the quantity that is induced is actually V .
This can be verified in the above experiment by
replacing the resistor with a high-impedance
voltmeter, which will indicate that V is changing even
though there is negligible current flow. Current flow
is simply a response to the induced potential.
Nevertheless, it is common to say informally that “I
is induced,” even if it is only indirectly through V .
So, if Lenz’s Law is simply an observation and not an
explanation of the underlying physics, then what is it
good for? Lenz’s Law is often useful for quickly
determining the direction of current flow in practical
electromagnetic induction problems, without
resorting to the mathematics associated with
Faraday’s Law. Here’s an example:
Example 8.1. Electromagnetic induction
through a transformer.
Figure 8.2 shows a rudimentary circuit
consisting of a battery and a switch on the left, a
voltmeter on the right, and a transformer linking
the two. It is not necessary to be familiar with
transformers to follow this example; suffice it to
say, the transformer considered here consists of
two coils wound around a common toroidal
core, which serves to contain magnetic flux. In
this way, the flux generated by either coil is
delivered to the other coil with negligible loss.
The experiment begins with the switch on the
left in the open state. Thus, there is no current
and no magnetic field apparent in the coil on the
left. The voltmeter reads 0 V. When the switch is
closed, what happens?
Solution. Closing the switch creates a current in
the coil on the left. Given the indicated polarity
of the battery, this current flows
counter-clockwise through the circuit on the left,
with current arriving in the left coil through the
bottom terminal. Given the indicated direction
of the winding in the left coil, the impressed
magnetic field Bimp is oriented
counter-clockwise through the toroidal core. The
coil on the right “sees” Bimp increase from zero
to some larger value. Since the voltmeter
presumably has input high impedance,
negligible current flows. However, if current
were able to flow, Lenz’s Law dictates that it
would be induced to flow in a counter-clockwise
direction around the circuit on the right, since
the induced magnetic field Bind would then be
clockwise-directed so as to oppose the increase
in Bimp . Therefore, the potential measured at
the bottom of the right coil would be higher than
the potential at the top of the right coil. The
8.2. ELECTROMAGNETIC INDUCTION
Magnet is ...
Motionless
Moving toward coil
Moving away from coil
177
|Bimp | in coil is ...
constant
increasing
decreasing
Circuit Response
V = 0, I = 0
V > 0, I > 0
V < 0, I < 0
Bind inside coil
none
Pointing right
Pointing left
Table 8.2: Results of the experiment associated with Figure 8.1.
negative feedback described by Lenz’s law.
+
-
-
+
-
V
Additional Reading:
+
• “Electromagnetic Induction” on Wikipedia.
• “Lenz’s Law” on Wikipedia.
E. Bach CC0 1.0 (modified)
Figure 8.2:
transformer.
Electromagnetic induction through a
figure indicates that the voltmeter measures the
potential at its right terminal relative to its left
terminal, so the needle will deflect to the right.
This deflection will be temporary, since the
current provided by the battery becomes
constant at a new non-zero value and Bind
responds only to the change in Bimp . The
voltmeter reading will remain at zero for as long
as the switch remains closed and the current
remains steady.
Here are some follow up exercises to test your
understanding of what is going on: (1) Now
open the switch. What happens? (2) Repeat the
original experiment, but before starting, swap
the terminals on the battery.
Finally, it is worth noting that Lenz’s law can also be
deduced from the principle of conservation of energy.
The argument is that if the induced magnetic field
reinforced the change in the impressed magnetic field,
then the sum magnetic field would increase. This
would result in a further increase in the induced
magnetic field, leading to a positive feedback
situation. However, positive feedback cannot be
supported without an external source of energy,
leading to a logical contradiction. In other words, the
principle of conservation of energy requires the
CHAPTER 8. TIME-VARYING FIELDS
178
8.3 Faraday’s Law
this single loop scenario:
VT = −
[m0055]
Faraday’s Law describes the generation of electric
potential by a time-varying magnetic flux. This is a
form of electromagnetic induction, as discussed in
Section 8.2.
To begin, consider the scenario shown in Figure 8.3.
A single loop of wire in the presence of an impressed
magnetic field B. For reasons explained later, we
introduce a small gap and define VT to be the
potential difference measured across the gap
according to the sign convention indicated. The
resistance R may be any value greater than zero,
including infinity; i.e., a literal gap.
As long as R is not infinite, we know from Lenz’s
Law (Section 8.2) to expect that a time-varying
magnetic field will cause a current to flow in the wire.
Lenz’s Law also tells us the direction in which the
current will flow. However, Lenz’s Law does not tell
us the magnitude of the current, and it sidesteps some
important physics that has profound implications for
the analysis and design of electrical devices,
including generators and transformers.
The more complete picture is given by Faraday’s Law.
In terms of the scenario of Figure 8.3, Faraday’s Law
relates the potential VT induced by the time variation
of B. VT then gives rise to the current identified in
Lenz’s Law. The magnitude of this current is simply
VT /R. Without further ado, here’s Faraday’s Law for
B
R
+
VT
-
Figure 8.3: A single loop of wire in the presence of
an impressed magnetic field.
∂
Φ (single loop)
∂t
(8.1)
Here Φ is the magnetic flux (units of Wb) associated
with any open surface S bounded by the loop:
Z
Φ=
B · ds
(8.2)
S
where B is magnetic flux density (units of T or
Wb/m2 ) and ds is the differential surface area vector.
To make headway with Faraday’s Law, one must be
clear about the meanings of S and ds. If the wire loop
in the present scenario lies in a plane, then a good
choice for S is the simply the planar area bounded by
the loop. However, any surface that is bounded by the
loop will work, including non-planar surfaces that
extend above and/or below the plane of the loop. All
that is required is that every magnetic field line that
passes through the loop also passes through S. This
happens automatically if the curve C defining the
edge of the open surface S corresponds to the loop.
Subsequently, the magnitude of ds is the differential
surface element ds and the direction of ds is the unit
vector n̂ perpendicular to each point on S, so
ds = n̂ds.
This leaves just one issue remaining – the orientation
of n̂. This is sorted out in Figure 8.4. There are two
possible ways for a vector to be perpendicular to a
surface, and the direction chosen for n̂ will affect the
sign of VT . Therefore, n̂ must be somehow related to
the polarity chosen for VT . Let’s consider this
relationship. Let C begin at the “−” terminal of VT
and follow the entire perimeter of the loop, ending at
the “+” terminal. Then, n̂ is determined by the
following “right-hand rule:” n̂ points in the direction
of the curled fingers of the right hand when the thumb
of the right hand is aligned in the direction of C. It’s
worth noting that this convention is precisely the
convention used to relate n̂ and C in Stokes’ Theorem
(Section 4.9).
Now let us recap how Faraday’s Law is applied to the
single loop scenario of Figure 8.3:
1. Assign “+” and “−” terminals to the gap voltage
VT .
8.3. FARADAY’S LAW
179
+
V
-
n
is distinct from the impressed magnetic field B and
which tends to oppose change in B. Thus, we see that
Faraday’s Law subsumes Lenz’s Law.
S
Frequently one is interested in a structure that consists
of multiple identical loops. We have in mind here
something like a coil with N ≥ 1 windings tightly
packed together. In this case, Faraday’s Law is
C
VT = −N
Figure 8.4: Relationship between the polarity of VT
and orientations of C and n̂ in the planar single-loop
scenario.
2. The orientation of n̂ is determined by the right
hand rule, taking the direction of C to be the
perimeter of the loop beginning at “−” and
ending at “+”
3. B yields a magnetic flux Φ associated with the
loop according to Equation 8.2. S is any open
surface that intersects all the magnetic field lines
that pass through the loop (so you might as well
choose S in a way that results in the simplest
possible integration).
4. By Faraday’s Law (Equation 8.1), VT is the time
derivative of Φ with a change of sign.
5. The current I flowing in the loop is VT /R, with
the reference direction (i.e., direction of positive
current) being from “+” to “−” through the
resistor. Think of the loop as a voltage source,
and you’ll get the correct reference direction for
I.
At this point, let us reiterate that the electromagnetic
induction described by Faraday’s Law induces
potential (in this case, VT ) and not current. The
current in the loop is simply the induced voltage
divided by the resistance of the loop. This point is
easily lost, especially in light of Lenz’s Law, which
seems to imply that current, as opposed to potential,
is the thing being induced.
Wondering about the significance of the minus sign in
Equation 8.1? That is specifically Lenz’s Law: The
current I that ends up circulating in the loop generates
its own magnetic field (“Bind ” in Section 8.2), which
∂
Φ
∂t
(8.3)
Note that the difference is simply that the gap
potential VT is greater by N . In other words, each
winding of the coil contributes a potential given by
Equation 8.1, and these potentials add in series.
Faraday’s Law, given in general by Equation 8.3,
states that the potential induced in a coil is proportional to the time derivative of the magnetic
flux through the coil.
The induced potential VT is often referred to as
“emf,” which is a contraction of the term
electromotive force – a misnomer to be sure, since no
actual force is implied, only potential. The term
“emf” is nevertheless frequently used in the context
of Faraday’s Law applications for historical reasons.
Previously in this section, we considered the
generation of emf by time variation of B. However,
Equation 8.1 indicates that what actually happens is
that the emf is the result of time variation of the
magnetic flux, Φ. Magnetic flux is magnetic flux
density integrated over area, so it appears that emf can
also be generated simply by varying S, independently
of any time variation of B. In other words, emf may
be generated even when B is constant, by instead
varying the shape or orientation of the coil. So, we
have a variety of schemes by which we can generate
emf. Here they are:
1. Time-varying B (as we considered previously in
this section). For example, B might be due to a
permanent magnet that is moved (e.g., translated
or rotated) in the vicinity of the coil, as described
in Section 8.2. Or, more commonly, B might be
due to a different coil that bears a time-varying
current. These mechanisms are collectively
CHAPTER 8. TIME-VARYING FIELDS
180
referred to as transformer emf. Transformer emf
is the underlying principle of operation of
transformers; for more on this see Section 8.5.
2. The perimeter C – and thus the surface S over
which Φ is determined – can be time-varying.
For example, a wire loop might be rotated or
changed in shape in the presence of a constant
magnetic field. This mechanism is referred to as
motional emf and is the underlying principle of
operation of generators (Section 8.7).
3. Transformer and motional emf can exist in
various combinations. From the perspective of
Faraday’s Law, transformer and motional emf
are the same in the sense in either case Φ is
time-varying, which is all that is required to
generate emf.
Finally, a comment on the generality of Faraday’s
Law. Above we have introduced Faraday’s Law as if
it were specific to loops and coils of wire. However,
the truth of the matter is that Faraday’s Law is
fundamental physics. If you can define a closed path
– current-bearing or not – then you can compute the
potential difference achieved by traversing that path
using Faraday’s Law. The value you compute is the
potential associated with electromagnetic induction,
and exists independently and in addition to the
potential difference associated with the static electric
field (e.g., Section 5.12). In other words:
Faraday’s Law indicates the contribution of electromagnetic induction (the generation of emf by
a time-varying magnetic flux) to the potential difference achieved by traversing a closed path.
In Section 8.8, this insight is used to transform the
static form of Kirchoff’s Voltage Law (Section 5.10)
– which gives the potential difference associated with
electric field only – into the Maxwell-Faraday
Equation (Section 8.8), which is a general statement
about the relationship between the instantaneous
value of the electric field and the time derivative of
the magnetic field.
8.4 Induction in a Motionless
Loop
[m0056]
In this section, we consider the problem depicted in
Figure 8.5, which is a single motionless loop of wire
in the presence of a spatially-uniform but
time-varying magnetic field. A small gap is
introduced in the loop, allowing us to measure the
induced potential VT . Additionally, a resistance R is
connected across VT in order to allow a current to
flow. This problem was considered in Section 8.3 as
an introduction to Faraday’s Law; in this section, we
shall actually work the problem and calculate some
values. This is intended to serve as an example of the
application of Faraday’s Law, a demonstration of
transformer emf, and will serve as a first step toward
an understanding of transformers as devices.
In the present problem, the loop is centered in the
z = 0 plane. The magnetic flux density is
B = b̂B(t); i.e., time-varying magnitude B(t) and a
constant direction b̂. Because this magnetic field is
spatially uniform (i.e., the same everywhere), we will
find that only the area of the loop is important, and
not it’s specific shape. For this reason, it will not be
necessary to specify the radius of the loop or even
require that it be a circular loop. Our task is to find
expressions for VT and I.
B
+
V
-
Additional Reading:
• “Faraday’s law of induction” on Wikipedia.
Figure 8.5: A single loop of wire in the presence of
an impressed spatially-uniform but time-varying magnetic field.
8.4. INDUCTION IN A MOTIONLESS LOOP
181
To begin, remember that Faraday’s Law is a
calculation of electric potential and not current. So,
the approach is to first find VT , and then find the
current I that flows through the gap resistance in
response.
The sign convention for VT is arbitrary; here, we have
selected “+” and “−” terminals as indicated in
Figure 8.5.1 Following the standard convention for
the reference direction of current through a passive
device, I should be directed as shown in Figure 8.5. It
is worth repeating that these conventions for the signs
of VT and I are merely references; for example, we
may well find that I is negative, which means that
current flows in a clockwise direction in the loop.
We now invoke Faraday’s Law:
∂
Φ
(8.4)
∂t
The number of windings N in the loop is 1, and Φ is
the magnetic flux through the loop. Thus:
Z
∂
B · ds
(8.5)
VT = −
∂t S
VT = −N
where S is any open surface that intersects all of the
magnetic field lines that pass through the loop. The
simplest such surface is simply the planar surface
defined by the perimeter of the loop. Then ds = n̂ds,
where ds is the differential surface element and n̂ is
the normal to the plane of the loop. Which of the two
possible normals to the loop? This is determined by
the right-hand rule of Stokes’ Theorem. From the
“−” terminal, we point the thumb of the right hand in
the direction that leads to the “+” terminal by
traversing the perimeter of the loop. When we do this,
the curled fingers of the right hand intersect S in the
same direction as n̂. To maintain the generality of
results derived below, we shall not make the
substitution n̂ = +ẑ; nevertheless we see this is the
case for a loop parallel to the z = 0 plane with the
polarity of VT indicated in Figure 8.5.
Taking this all into account, we have
Z ∂
VT = −
b̂B(t) · (n̂ds)
∂t S
∂ Z
= − b̂ · n̂
B(t) ds
∂t S
Since the magnetic field is uniform, B(t) may be
extracted from the integral. Furthermore, the shape
and the orientation of the loop are time-invariant, so
the remaining integral may be extracted from the time
derivative operation. This leaves:
Z
∂
B(t)
ds
(8.7)
VT = − b̂ · n̂
∂t
S
The integral in this expression is simply the area of
the loop, which is a constant; let the symbol A
represent this area. We obtain
∂
VT = − b̂ · n̂A
B(t)
∂t
(8.8)
which is the expression we seek. Note that the
quantity b̂ · n̂A is the projected area of the loop. The
projected area is equal to A when the the magnetic
field lines are perpendicular to the loop (i.e., b̂ = n̂),
and decreases to zero as b̂ · n̂ → 0. Summarizing:
The magnitude of the transformer emf induced
by a spatially-uniform magnetic field is equal to
the projected area times the time rate of change of
the magnetic flux density, with a change of sign.
(Equation 8.8).
A few observations about this result:
• As promised earlier, we have found that the
shape of the loop is irrelevant; i.e., a square loop
having the same area and planar orientation
would result in the same VT . This is because the
magnetic field is spatially uniform, and because
it is the magnetic flux (Φ) and not the magnetic
field or shape of the loop alone that determines
the induced potential.
• The induced potential is proportional to A; i.e.,
VT can be increased by increasing the area of the
loop.
• The peak magnitude of the induced potential is
maximized when the plane of the loop is
perpendicular to the magnetic field lines.
(8.6)
1 A good exercise for the student is to repeat this problem with
the terminal polarity reversed; one should obtain the same answer.
• The induced potential goes to zero when the
plane of the loop is parallel to the magnetic field
lines. Said another way, there is no induction
unless magnetic field lines pass through the loop.
CHAPTER 8. TIME-VARYING FIELDS
182
• The induced potential is proportional to the rate
of change of B. If B is constant in time, then
there is no induction.
Finally, the current in the loop is simply
I=
VT
R
(8.9)
Again, electromagnetic induction induces potential,
and the current flows only in response to the induced
potential as determined by Ohm’s Law. In particular,
if the resistor is removed, then R → ∞ and I → 0,
but VT is unchanged.
One final comment is that even though the current I is
not a direct result of electromagnetic induction, we
can use I as a check of the result using Lenz’s Law
(Section 8.2). We’ll demonstrate this in the example
below.
Example 8.2. Induction in a motionless circular
loop by a linearly-increasing magnetic field
Let the loop be planar in the z = 0 plane and
circular with radius a = 10 cm. Let the
magnetic field be ẑB(t) where
B(t) = 0
,t<0
= B0 t/t0 , 0 ≤ t ≤ t0
= B0
, t > t0
(8.10)
i.e., B(t) begins at zero and increases linearly to
B0 at time t0 , after which it remains constant at
B0 . Let B0 = 0.2 T, t0 = 3 s, and let the loop be
closed by a resistor R = 1 kΩ. What current I
flows in the loop?
Solution. Adopting the sign conventions of
Figure 8.5 we first note that n̂ = +ẑ; this is
determined by the right-hand rule with respect to
the indicated polarity of VT . Thus, Equation 8.8
becomes
∂
VT = − b̂ · ẑA
B(t)
∂t
Note b̂ · ẑA = A since b̂ = ẑ; i.e., because the
plane of the loop is perpendicular to the
magnetic field lines. Since the loop is circular,
A = πa2 . Also
∂
B(t) = 0
,t<0
∂t
= B0 /t0 , 0 ≤ t ≤ t0
=0
, t > t0
Putting this all together:
VT = −πa2
B0
= −2.09 mV , 0 ≤ t ≤ t0
t0
and VT = 0 before and after this time interval,
since B is constant during those times.
Subsequently the induced current is
I=
VT
= −2.09 µA , 0 ≤ t ≤ t0
R
and I = 0 before and after this time interval. We
have found that the induced current is a constant
clockwise flow that exists only while B is
increasing.
Finally, let’s see if the result is consistent with
Lenz’s Law. The current induced while B is
changing gives rise to an induced magnetic field
Bind . From the right-hand rule that relates the
direction of I to the direction of Bind
(Section 7.5), the direction of Bind is generally
−ẑ inside the loop. In other words, the magnetic
field associated with the induced current opposes
the increasing impressed magnetic field that
induced the current, in accordance with Lenz’s
Law.
Example 8.3. Induction in a motionless circular
loop by a sinusoidally-varying magnetic field.
Let us repeat the previous example, but now with
B(t) = B0 sin 2πf0 t
with f0 = 1 kHz.
Solution. Now
∂
B(t) = 2πf0 B0 cos 2πf0 t
∂t
8.5. TRANSFORMERS: PRINCIPLE OF OPERATION
So
VT = −2π 2 f0 a2 B0 cos 2πf0 t
183
8.5 Transformers: Principle of
Operation
Subsequently
I=
2π 2 f0 a2 B0
VT
=−
cos 2πf0 t
R
R
Substituting values, we have:
I = −(39.5 mA) cos [(6.28 krad/s)t]
It should be no surprise that VT and I vary
sinusoidally, since the source (B) varies
sinusoidally. A bit of useful trivia here is that VT
and I are 90◦ out of phase with the source. It is
also worth noting what happens when B(t) = 0.
This occurs twice per period, at t = n/2f0
where n is any integer, including t = 0. At these
times B(t) is zero, but VT and hence IR are
decidedly non-zero; in fact, they are at their
maximum magnitude. Again, it is the change in
B that induces voltage and subsequently current,
not B itself.
[m0031]
A transformer is a device that connects two electrical
circuits through a shared magnetic field.
Transformers are used in impedance transformation,
voltage level conversion, circuit isolation, conversion
between single-ended and differential signal modes,
and other applications.2 The underlying
electromagnetic principle is Faraday’s Law
(Section 8.3) – in particular, transformer emf.
The essential features of a transformer can be derived
from the simple experiment shown in Figures 8.6 and
8.7. In this experiment, two coils are arranged along a
common axis. The winding pitch is small, so that all
magnetic field lines pass through the length of the
coil, and no lines pass between the windings. To
further contain the magnetic field, we assume both
coils are wound on the same core, consisting of some
material exhibiting high permeability. The upper coil
has N1 turns and the lower coil has N2 turns.
In Part I of this experiment (Figure 8.6), the upper
coil is connected to a sinusoidally-varying voltage
(1)
source V1 in which the subscript refers to the coil
and the superscript refers to “Part I” of this
experiment. The voltage source creates a current in
the coil, which in turn creates a time-varying
magnetic field B in the core.
The lower coil has N2 turns wound in the opposite
direction and is open-circuited. Given the
closely-spaced windings and use of a
high-permeability core, we assume that the magnetic
field within the lower coil is equal to B generated in
the upper coil. The potential induced in the lower coil
(1)
is V2 with reference polarity indicated in the figure.
From Faraday’s Law we have
(1)
V2
= −N2
∂
Φ2
∂t
(8.11)
where Φ2 is the flux through a single turn of the lower
2 See “Additional Reading” at the end of this section for more on
these applications.
CHAPTER 8. TIME-VARYING FIELDS
184
coil. Thus:
∂
(1)
V2 = −N2
∂t
+
V 1(1)
N1 turns
−
+
V 2(1)
Z
S
B · (−ẑds)
(8.12)
Note that the direction of ds = −ẑds is determined
(1)
by the polarity we have chosen for V2 .
In Part II of the experiment (Figure 8.7), we make
(2)
changes as follows. We apply a voltage V2 to the
lower coil and open-circuit the upper coil. Further, we
(2)
adjust V2 such that the induced magnetic flux
density is again B – that is, equal to the field in Part I
of the experiment. Now we have
N2 turns
(2)
V1
−
∂
Φ1
∂t
(8.13)
B · (+ẑds)
(8.14)
= −N1
which is
Figure 8.6: Part I of an experiment demonstrating the
linking of electric circuits using a transformer.
∂
(2)
V1 = −N1
∂t
Z
S
For reasons that will become apparent in a moment,
let’s shift the leading minus sign into the integral. We
then have
Z
∂
(2)
V1 = +N1
B · (−ẑds)
(8.15)
∂t S
Comparing this to Equations 8.11 and 8.12, we see
that we may rewrite this in terms of the flux in the
lower coil in Part I of the experiment:
+
V (2)
N1 turns
−
+
V 2(2)
N2 turns
−
Figure 8.7: Part II of an experiment demonstrating the
linking of electric circuits using a transformer.
(2)
V1
= +N1
∂
Φ2
∂t
(8.16)
In fact, we can express this in terms of the potential in
Part I of the experiment:
N1
∂
(2)
V1 = −
−N2 Φ2
N2
∂t
N1
(1)
= −
V2
(8.17)
N2
We have found that the potential in the upper coil in
Part II is related in a simple way to the potential in the
lower coil in Part I of the experiment. If we had done
Part II first, we would have obtained the same result
but with the superscripts swapped. Therefore, it must
be generally true – regardless of the arrangement of
terminations – that
V1 = −
N1
V2
N2
(8.18)
8.6. TRANSFORMERS AS TWO-PORT DEVICES
This expression should be familiar from elementary
circuit theory – except possibly for the minus sign.
The minus sign is a consequence of the fact that the
coils are wound in opposite directions. We can make
the above expression a little more general as follows:
V1
N1
=p
V2
N2
185
Magnetic
Flux, Φ
−
N 1 turns
−
(8.19)
where p is defined to +1 when the coils are wound in
the same direction and −1 when coils are wound in
opposite directions. (It is an excellent exercise to
confirm that this is true by repeating the above
analysis with winding direction changed for either the
upper or lower coil, for which p will then turn out to
be +1.) This is the “transformer law” of basic electric
circuit theory, from which all other characteristics of
transformers as two-port circuit devices can be
obtained (See Section 8.6 for follow-up on that).
Summarizing:
The ratio of coil voltages in an ideal transformer
is equal to the ratio of turns with sign determined
by the relative directions of the windings, per
Equation 8.19.
V1
+
V2
N2 turns
+
Transform
er
Core
c BillC (modified) CC BY SA 3.0
Figure 8.8: Transformer implemented as coils sharing
a toroidal core. Here p = +1.
8.6 Transformers as Two-Port
Devices
[m0032]
A more familiar transformer design is shown in
Figure 8.8 – coils wound on a toroidal core as
opposed to a cylindrical core. Why do this? This
arrangement confines the magnetic field linking the
two coils to the core, as opposed to allowing field
lines to extend beyond the device. This confinement
is important in order to prevent fields originating
outside the transformer from interfering with the
magnetic field linking the coils, which would lead to
electromagnetic interference (EMI) and
electromagnetic compatibility (EMC) problems. The
principle of operation is in all other respects the same.
Additional Reading:
• “Transformer” on Wikipedia.
• “Balun” on Wikipedia.
Section 8.5 explains the principle of operation of the
ideal transformer. The relationship governing the
terminal voltages V1 and V2 was found to be
N1
V1
=p
V2
N2
(8.20)
where N1 and N2 are the number of turns in the
associated coils and p is either +1 or −1 depending
on the relative orientation of the windings; i.e.,
whether the reference direction of the associated
fluxes is the same or opposite, respectively.
We shall now consider ratios of current and
impedance in ideal transformers, using the two-port
model shown in Figure 8.9. By virtue of the reference
current directions and polarities chosen in this figure,
the power delivered by the source V1 is V1 I1 , and the
power absorbed by the load Z2 is −V2 I2 . Assuming
the transformer windings have no resistance, and
assuming the magnetic flux is perfectly contained
within the core, the power absorbed by the load must
equal the power provided by the source; i.e.,
CHAPTER 8. TIME-VARYING FIELDS
186
V1 I1 = −V2 I2 . Thus, we have3
V2
N2
I1
=−
= −p
I2
V1
N1
(8.21)
We can develop an impedance relationship for ideal
transformers as follows. Let Z1 be the input
impedance of the transformer; that is, the impedance
looking into port 1 from the source. Thus, we have
V1
I1
+p (N1 /N2 ) V2
=
−p (N2 /N1 ) I2
2 V2
N1
=−
N2
I2
Z1 ,
Z1
=
Z2
N1
N2
2
(8.22)
(8.24)
3 The minus signs in this equation are a result of the reference
polarity and directions indicated in Figure 8.9. These are more-orless standard in electrical two-port theory (see “Additional Reading”
at the end of this section), but are certainly not the only reasonable choice. If you see these expressions without the minus signs,
it probably means that a different combination of reference directions/polarities is in effect.
I1
+
_
I2
+
+
V1
V
−
−
The relationships developed above should be viewed
as AC expressions, and are not normally valid at DC.
This is because transformers exhibit a fundamental
limitation in their low-frequency performance. To see
this, first recall Faraday’s Law:
V = −N
In Figure 8.9, Z2 is the the output impedance of port
2; that is, the impedance looking out port 2 into the
load. Therefore, Z2 = −V2 /I2 (once again the minus
sign is a result of our choice of sign conventions in
Figure 8.9). Substitution of this result into the above
equation yields
2
N1
Z1 =
Z2
(8.23)
N2
and therefore
Thus, we have demonstrated that a transformer scales
impedance in proportion to the square of the turns
ratio N1 /N2 . Remarkably, the impedance
transformation depends only on the turns ratio, and is
independent of the relative direction of the windings
(p).
Z2
∂
Φ
∂t
(8.25)
If the magnetic flux Φ isn’t time-varying, then there is
no induced electric potential, and subsequently no
linking of the signals associated with the coils. At
very low but non-zero frequencies, we encounter
another problem that gets in the way – magnetic
saturation. To see this, note we can obtain an
expression for Φ from Faraday’s Law by integrating
with respect to time, yielding
Φ(t) = −
1
N
Z t
V (τ )dτ + Φ(t0 )
(8.26)
t0
where t0 is some earlier time at which we know the
value of Φ(t0 ). Let us assume that V (t) is
sinusoidally-varying. Then the peak value of Φ after
t = t0 depends on the frequency of V (t). If the
frequency of V (t) is very low, then Φ can become
very large. Since the associated cross-sectional areas
of the coils are presumably constant, this means that
the magnetic field becomes very large. The problem
with that is that most practical high-permeability
materials suitable for use as transformer cores exhibit
magnetic saturation; that is, the rate at which the
magnetic field is able to increase decreases with
increasing magnetic field magnitude (see
Section 7.16). The result of all this is that a
transformer may work fine at (say) 1 MHz, but at
(say) 1 Hz the transformer may exhibit an apparent
loss associated with this saturation. Thus, practical
transformers exhibit highpass frequency response.
Z1
Figure 8.9: The transformer as a two-port circuit device.
It should be noted that the highpass behavior of
practical transistors can be useful. For example, a
transformer can be used to isolate an undesired DC
offset and/or low-frequency noise in the circuit
8.7. THE ELECTRIC GENERATOR
SingleEnded
Differential
187
SingleEnded
8.7 The Electric Generator
[m0030]
c SpinningSpark CC BY SA 3.0 (modified)
Figure 8.10: Transformers used to convert a singleended (“unbalanced”) signal to a differential (balanced) signal, and back.
attached to one coil from the circuit attached to the
other coil.
The DC-isolating behavior of a transformer also
allows the transformer to be used as a balun, as
illustrated in Figure 8.10. A balun is a two-port
device that transforms a single-ended (“unbalanced”)
signal – that is, one having an explicit connection to a
datum (e.g., ground) – into a differential (“balanced”)
signal, for which there is no explicit connection to a
datum. Differential signals have many benefits in
circuit design, whereas inputs and outputs to devices
must often be in single-ended form. Thus, a common
use of transformers is to effect the conversion
between single-ended and differential circuits.
Although a transformer is certainly not the only
device that can be used as a balun, it has one very big
advantage, namely bandwidth.
Additional Reading:
• “Transformer” on Wikipedia.
• “Two-port network” on Wikipedia.
• “Saturation (magnetic)” on Wikipedia.
• “Balun” on Wikipedia.
• S.W. Ellingson, “Differential Circuits” (Sec. 8.7)
in Radio Systems Engineering, Cambridge Univ.
Press, 2016.
A generator is a device that transforms mechanical
energy into electrical energy, typically by
electromagnetic induction via Faraday’s Law
(Section 8.3). For example, a generator might consist
of a gasoline engine that turns a crankshaft to which
is attached a system of coils and/or magnets. This
rotation changes the relative orientations of the coils
with respect to the magnetic field in a time-varying
manner, resulting in a time-varying magnetic flux and
subsequently induced electric potential. In this case,
the induced potential is said to be a form of motional
emf, as it is due entirely to changes in geometry as
opposed to changes in the magnitude of the magnetic
field. Coal- and steam-fired generators, hydroelectric
generators, wind turbines, and other energy
generation devices operate using essentially this
principle.
Figure 8.11 shows a rudimentary generator, which
serves as to illustrate the relevant points. This
generator consists of a planar loop that rotates around
the z axis; therefore, the rotation can be parameterized
in φ. In this case, the direction of rotation is specified
to be in the +φ direction. The frequency of rotation is
f0 ; that is, the time required for the loop to make one
complete revolution is 1/f0 . We assume a
time-invariant and spatially-uniform magnetic flux
density B = b̂B0 where b̂ and B0 are both constants.
For illustration purposes, the loop is indicated to be
circular. However, because the magnetic field is
time-invariant and spatially-uniform, the specific
shape of the loop is not important, as we shall see in a
moment. Only the area of the loop will matter.
The induced potential is indicated as VT with
reference polarity as indicated in the figure. This
potential is given by Faraday’s Law:
∂
Φ
(8.27)
∂t
Here Φ is the magnetic flux associated with an open
surface S bounded by the loop:
Z
Φ=
B · ds
(8.28)
VT = −
S
As usual, S can be any surface that intersects all
CHAPTER 8. TIME-VARYING FIELDS
188
z
y
n̂(t) = −x̂ sin 2πf0 t + ŷ cos 2πf0 t
(8.30)
as can be confirmed by checking the three special
cases identified above. Now applying Faraday’s Law,
we find
+V
-
see that at t = 0, n̂ = +ŷ. A bit later, the loop will
have rotated by one-fourth of a complete rotation, and
at that time n̂ = −x̂. This occurs at t = 1/(4f0 ).
Later still, the loop will have rotated by one-half of a
compete rotation, and then n̂ = −ŷ. This occurs at
t = 1/(2f0 ). It is apparent that
∂
Φ
∂t
i
∂ h
=−
b̂ · n̂(t) AB0
∂t
∂
= − b̂ · n̂(t) AB0
∂t
VT = −
x
direction of
rotation
Figure 8.11: A rudimentary single-loop generator,
shown at time t = 0.
magnetic field lines passing through the loop, and
also as usual, the simplest choice is simply the planar
area bounded by the loop. The differential surface
element ds is n̂ds, where n̂ is determined by the
reference polarity of VT according to the “right hand
rule” convention from Stokes’ Theorem. Making
substitutions, we have
Z b̂B0 · (n̂ds)
Φ=
S
Z
h
i
= b̂ · n̂ B0
ds
h
i S
= b̂ · n̂A B0
(8.29)
where A is the area of the loop.
To make headway, an expression for n̂ is needed. The
principal difficulty here is that n̂ is rotating with the
loop, so it is time-varying. To sort this out, first
consider the situation at time t = 0, which is the
moment illustrated in Figure 8.11. Beginning at the
“−” terminal, we point the thumb of our right hand in
the direction that leads to the “+” terminal by
traversing the loop; n̂ is then the direction
perpendicular to the plane of the loop in which the
fingers of our right hand pass through the loop. We
(8.31)
For notational convenience we make the following
definition:
1 ∂
n̂′ (t) , −
n̂(t)
(8.32)
2πf0 ∂t
which is simply the time derivative of n̂ divided by
2πf0 so as to retain a unit vector. The reason for
including a change of sign will become apparent in a
moment. Applying this definition, we find
n̂′ (t) = +x̂ cos 2πf0 t + ŷ sin 2πf0 t
(8.33)
Note that this is essentially the definition of the radial
basis vector ρ̂ from the cylindrical coordinate system
(which is why we applied the minus sign in
Equation 8.32). Recognizing this, we write
ρ̂(t) = +x̂ cos 2πf0 t + ŷ sin 2πf0 t
(8.34)
and finally
VT = +2πf0 AB0 b̂ · ρ̂(t)
(8.35)
If the purpose of this device is to generate power, then
presumably we would choose the magnetic field to be
in a direction that maximizes the maximum value of
b̂ · ρ̂(t). Therefore, power is optimized for B
polarized entirely in some combination of x̂ and ŷ,
and with B · ẑ = 0. Under that constraint, we see that
VT varies sinusoidally with frequency f0 and exhibits
peak magnitude
max |VT (t)| = 2πf0 AB0
(8.36)
8.8. THE MAXWELL-FARADAY EQUATION
It’s worth noting that the maximum voltage
magnitude is achieved when the plane of the loop is
parallel to B; i.e., when b̂ · n̂(t) = 0 so that
Φ(t) = 0. Why is that? Because this is when Φ(t) is
most rapidly increasing or decreasing. Conversely,
when the plane of the loop is perpendicular to B (i.e.,
b̂ · n̂(t) = 1), |Φ(t)| is maximum but its
time-derivative is zero, so VT (t) = 0 at this instant.
Example 8.4. Rudimentary electric generator.
The generator in Figure 8.11 consists of a
circular loop of radius a = 1 cm rotating at 1000
revolutions per second in a static and
spatially-uniform magnetic flux density of 1 mT
in the +x̂ direction. What is the induced
potential?
Solution. From the problem statement,
f0 = 1 kHz, B0 = 1 mT, and b̂ = +x̂.
Therefore b̂ · ρ̂(t) = x̂ · ρ̂(t) = cos 2πf0 t. The
area of the loop is A = πa2 . From
Equation 8.35 we obtain
VT (t) ∼
= (1.97 mV) cos [(6.28 krad/s) t]
Finally, we note that it is not really necessary for the
loop to rotate in the presence of a magnetic field with
constant b̂; it works equally well for the loop to be
stationary and for b̂ to rotate – in fact, this is
essentially the same problem. In some practical
generators, both the potential-generating coils and
fields (generated by some combination of magnets
and coils) rotate.
Additional Reading:
• “Electric Generator” on Wikipedia.
189
8.8 The Maxwell-Faraday
Equation
[m0050]
In this section, we generalize Kirchoff’s Voltage Law
(KVL), previously encountered as a principle of
electrostatics in Sections 5.10 and 5.11. KVL states
that in the absence of a time-varying magnetic flux,
the electric potential accumulated by traversing a
closed path C is zero. Here is that idea in
mathematical form:
I
V =
E · dl = 0
(8.37)
C
Now recall Faraday’s Law (Section 8.3):
Z
∂
∂
B · ds
V =− Φ=−
∂t
∂t S
(8.38)
Here, S is any open surface that intersects all
magnetic field lines passing through C, with the
relative orientations of C and ds determined in the
usual way by the Stokes’ Theorem convention
(Section 4.9). Note that Faraday’s Law agrees with
KVL in the magnetostatic case. If magnetic flux is
constant, then Faraday’s Law says V = 0. However,
Faraday’s Law is very clearly not consistent with
KVL if magnetic flux is time-varying. The correction
is simple enough; we can simply set these expressions
to be equal. Here we go:
I
C
E · dl = −
∂
∂t
Z
S
B · ds
(8.39)
This general form is known by a variety of names;
here we refer to it as the Maxwell-Faraday Equation
(MFE).
The integral form of the Maxwell-Faraday Equation (Equation 8.39) states that the electric potential associated with a closed path C is due entirely
to electromagnetic induction, via Faraday’s Law.
Despite the great significance of this expression as
one of Maxwell’s Equations, one might argue that all
we have done is simply to write Faraday’s Law in a
slightly more verbose way. This is true. The real
power of the MFE is unleashed when it is expressed
CHAPTER 8. TIME-VARYING FIELDS
190
in differential, as opposed to integral form. Let us
now do this.
We can transform the left-hand side of Equation 8.39
into a integral over S using Stokes’ Theorem.
Applying Stokes’ theorem on the left, we obtain
Z
Z
∂
B · ds
(8.40)
(∇ × E) · ds = −
∂t S
S
Now exchanging the order of integration and
differentiation on the right hand side:
Z
Z ∂
(∇ × E) · ds =
− B · ds
∂t
S
S
(8.41)
∂
B
∂t
(8.42)
which is the MFE in differential form.
What does this mean? Recall that the curl of E is a
way to take a directive of E with respect to position
(Section 4.8). Therefore the MFE constrains spatial
derivatives of E to be simply related to the rate of
change of B. Said plainly:
The differential form of the Maxwell-Faraday
Equation (Equation 8.42) relates the change in
the electric field with position to the change in
the magnetic field with time.
Now that is arguably new and useful information. We
now see that electric and magnetic fields are coupled
not only for line integrals and fluxes, but also at each
point in space.
Additional Reading:
• “Faraday’s Law of Induction” on Wikipedia.
• “Maxwell’s Equations” on Wikipedia.
[m0053]
In this section, we generalize Ampere’s Law,
previously encountered as a principle of
magnetostatics in Sections 7.4 and 7.9. Ampere’s
Law states that the current Iencl flowing through
closed path C is equal to the line integral of the
magnetic field intensity H along C. That is:
I
H · dl = Iencl
(8.43)
C
The surface S on both sides is the same, and we have
not constrained S in any way. S can be any
mathematically-valid open surface anywhere in space,
having any size and any orientation. The only way the
above expression can be universally true under these
conditions is if the integrands on each side are equal
at every point in space. Therefore,
∇×E=−
8.9 Displacement Current and
Ampere’s Law
We shall now demonstrate that this equation is
unreliable if the current is not steady; i.e., not DC.
First, consider the situation shown in Figure 8.12.
Here, a current I flows in the wire, subsequently
generating a magnetic field H that circulates around
the wire (Section 7.5). When we perform the
integration in Ampere’s Law along any path C
enclosing the wire, the result is I, as expected. In this
case, Ampere’s Law is working even when I is
time-varying.
Now consider the situation shown in Figure 8.13, in
which we have introduced a parallel-plate capacitor.
In the DC case, this situation is simple. No current
flows, so there is no magnetic field and Ampere’s Law
is trivially true. In the AC case, the current I can be
non-zero, but we must be clear about the physical
origin of this current. What is happening is that for
one half of a period, a source elsewhere in the circuit
is moving positive charge to one side of the capacitor
and negative charge to the other side. For the other
half-period, the source is exchanging the charge, so
that negative charge appears on the previously
positively-charged side and vice-versa. Note that at
no point is current flowing directly from one side of
the capacitor to the other; instead, all current must
flow through the circuit in order to arrive at the other
plate. Even though there is no current between the
plates, there is current in the wire, and therefore there
is also a magnetic field associated with that current.
Now we are ready to shine a light on the problem.
Recall that from Stokes’ Theorem (Section 4.9), the
line integral over C is mathematically equivalent to an
8.9. DISPLACEMENT CURRENT AND AMPERE’S LAW
I
H
C
S1
191
integral over any open surface S that is bounded by C.
Two such surfaces are shown in Figure 8.12 and
Figure 8.13, indicated as S1 and S2 . In the wire-only
scenario of Figure 8.12, the choice of S clearly
doesn’t matter; any valid surface intersects current
equal to I. Similarly in the scenario of Figure 8.13,
everything seems fine if we choose S = S1 . If, on the
other hand, we select S2 in the parallel-plate capacitor
case, then we have a problem. There is no current
flowing through S2 , so the right side of Equation 8.43
is zero even though the left side is potentially
non-zero. So, it appears that something necessary for
the time-varying case is missing from Equation 8.43.
To resolve the problem, we postulate an additional
term in Ampere’s Law that is non-zero in the above
scenario. Specifically, we propose:
I
H · dl = Ic + Id
(8.44)
I
C
C. Burks (modified)
Figure 8.12: Ampere’s Law applied to a continuous
line of current.
where Ic is the enclosed current (formerly identified
as Iencl ) and Id is the proposed new term. If we are to
accept this postulate, then here is a list of things we
know about Id :
• Id has units of current (A).
I
H
C
S1
• Id = 0 in the DC case and is potentially
non-zero in the AC case. This implies that Id is
the time derivative of some other quantity.
• Id must be somehow related to the electric field.
E
I
C. Burks (modified)
Figure 8.13: Ampere’s Law applied to a parallel plate
capacitor.
How do we know Id must be related to the electric
field? This is because the Maxwell-Faraday Equation
(Section 8.8) tells us that spatial derivatives of E are
related to time derivatives of H; i.e., E and H are
coupled in the time-varying (here, AC) case. This
coupling between E and H must also be at work here,
but we have not yet seen E play a role. This is pretty
strong evidence that Id depends on the electric field.
Without further ado, here’s Id :
Z
∂D
· ds
Id =
S ∂t
(8.45)
where D is the electric flux density (units of C/m2 )
and is equal to ǫE as usual, and S is the same open
surface associated with C in Ampere’s Law. Note that
CHAPTER 8. TIME-VARYING FIELDS
192
this expression meets our expectations: It is
determined by the electric field, it is zero when the
electric field is constant (i.e., not time varying), and
has units of current.
The quantity Id is commonly known as displacement
current. It should be noted that this name is a bit
misleading, since Id is not a current in the
conventional sense. Certainly, it is not a conduction
current – conduction current is represented by Ic , and
there is no current conducted through an ideal
capacitor. It is not unreasonable to think of Id as
current in a more general sense, for the following
reason. At one instant, charge is distributed one way
and at another, it is distributed in another way. If you
define current as a time variation in the charge
distribution relative to S – regardless of the path taken
by the charge – then Id is a current. However, this
distinction is a bit philosophical, so it may be less
confusing to interpret “displacement current” instead
as a separate electromagnetic quantity that just
happens to have units of current.
Now we are able to write the general form of
Ampere’s Law that applies even when sources are
time-varying. Here it is:
I
Z
∂D
H · dl = Ic +
· ds
(8.46)
C
S ∂t
As is the case in the Maxwell-Faraday Equation, most
of the utility of Ampere’s Law is unleashed when
expressed in differential form. To obtain this form the
first step is to write Ic as an integral of over S; this is
simply (see Section 6.2):
Z
Ic =
J · ds
(8.47)
S
where J is the volume current density (units of
A/m2 ). So now we have
I
Z
Z
∂D
· ds
H · dl =
J · ds +
C
S ∂t
ZS ∂D
· ds
(8.48)
=
J+
∂t
S
We can transform the left side of the above equation
into a integral over S using Stokes’ Theorem
(Section 4.9). We obtain
Z
Z ∂D
(∇ × H) · ds =
J+
· ds (8.49)
∂t
S
S
The surface S on both sides is the same, and we have
not constrained S in any way. S can be any
mathematically-valid open surface anywhere in space,
having any size and any orientation. The only way the
above expression can be universally true under these
conditions is if the integrands on each side are equal
at every point in space. Therefore:
∇×H=J+
∂
D
∂t
(8.50)
which is Ampere’s Law in differential form.
What does Equation 8.50 mean? Recall that the curl
of H is a way to describe the direction and rate of
change of H with position. Therefore, this equation
constrains spatial derivatives of H to be simply
related to J and the time derivative of D
(displacement current). Said plainly:
The differential form of the general (timevarying) form of Ampere’s Law (Equation 8.50)
relates the change in the magnetic field with position to the change in the electric field with time,
plus current.
As is the case in the Maxwell-Faraday Equation
(Section 8.8), we see that electric and magnetic fields
become coupled at each point in space when sources
are time-varying.
Additional Reading:
• “Displacement Current” on Wikipedia.
• “Ampere’s Circuital Law” on Wikipedia.
• “Maxwell’s Equations” on Wikipedia.
[m0126]
8.9. DISPLACEMENT CURRENT AND AMPERE’S LAW
193
Image Credits
Fig. 8.1: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fCoilBarMagnet.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/), modified from the original.
Fig. 8.2: E. Bach, https://commons.wikimedia.org/wiki/File:Faraday emf experiment.svg,
public domain via CC0 1.0 (https://creativecommons.org/publicdomain/zero/1.0/deed.en), modified from
the original.
Fig. 8.8: c BillC, https://commons.wikimedia.org/wiki/File:Transformer3d col3.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Modified by author.
Fig. 8.10: c SpinningSpark, https://en.wikipedia.org/wiki/File:Transformer balance.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Modified by author.
Fig. 8.12: C. Burks, https://en.wikipedia.org/wiki/File:Displacement current in capacitor.svg,
public domain. Heavily modified by author.
Fig. 8.13: C. Burks, https://en.wikipedia.org/wiki/File:Displacement current in capacitor.svg,
public domain. Heavily modified by author.
Chapter 9
Plane Waves in Lossless Media
9.1 Maxwell’s Equations in
Differential Phasor Form
and
[m0042]
In this section, we derive the phasor form of
Maxwell’s Equations from the general time-varying
form of these equations. Here we are interested
exclusively in the differential (“point”) form of these
equations. It is assumed that the reader is comfortable
with phasor representation and its benefits; if not, a
review of Section 1.5 is recommended before
attempting this section.
Maxwell’s Equations in differential time-domain
form are Gauss’ Law (Section 5.7):
∇ · D = ρv
(9.1)
the Maxwell-Faraday Equation (MFE; Section 8.8):
∇×E=−
∂
B
∂t
(9.2)
Gauss’ Law for Magnetism (GSM; Section 7.3):
∇·B=0
(9.3)
and Ampere’s Law (Section 8.9):
∇×H=J+
∂
D
∂t
(9.4)
We begin with Gauss’s Law (Equation 9.1). We
e and ρev as phasor quantities through the
define D
usual relationship:
n
o
e jωt
D = Re De
(9.5)
ρv = Re ρev ejωt
(9.6)
Substituting these expressions into Equation 9.1:
h n
oi
e jωt = Re ρev ejωt
∇ · Re De
(9.7)
Divergence is a real-valued linear operator. Therefore,
we may exchange the order of the “Re” and “∇·”
operations (this is “Claim 2” from Section 1.5):
n
h
io
e jωt = Re ρev ejωt
Re ∇ · De
(9.8)
Next, we note that the differentiation associated with
the divergence operator is with respect to position and
not with respect to time, so the order of operations
may be further rearranged as follows:
o
nh
i
e ejωt = Re ρev ejωt
(9.9)
Re ∇ · D
Finally, we note that the equality of the left and right
sides of the above equation implies the equality of the
associated phasors (this is “Claim 1” from
Section 1.5); thus,
e = ρev
∇·D
(9.10)
In other words, the differential form of Gauss’ Law
for phasors is identical to the differential form of
Gauss’ Law for physical time-domain quantities.
The same procedure applied to the MFE is only a
little more complicated. First, we establish the phasor
representations of the electric and magnetic fields:
n
o
e jωt
E = Re Ee
(9.11)
n
o
e jωt
B = Re Be
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
(9.12)
9.1. MAXWELL’S EQUATIONS IN DIFFERENTIAL PHASOR FORM
After substitution into Equation 9.2:
h n
oi
h n
oi
e jωt = − ∂ Re Be
e jωt
∇ × Re Ee
(9.13)
∂t
Both curl and time-differentiation are real-valued
linear operations, so we are entitled to change the
order of operations as follows:
h
n
h
io
∂ e jωt i
jωt
e
Re ∇ × Ee
= −Re
(9.14)
Be
∂t
On the left, we note that the time dependence ejωt can
be pulled out of the argument of the curl operator,
since it does not depend on position:
h
o
nh
i
∂ e jωt i
jωt
e
= −Re
Re ∇ × E e
(9.15)
Be
∂t
e is constant with respect
On the right, we note that B
to time (because it is a phasor), so:
o
nh
i
∂ jωt
jωt
e
e
= −Re B e
Re ∇ × E e
∂t
n
o
jωt
e
= −Re Bjωe
o
nh
i
e ejωt (9.16)
= Re −jω B
And so we have found:
e = −jω B
e
∇×E
(9.17)
Let’s pause for a moment to consider the above result.
In the general time-domain version of the MFE, we
must take spatial derivatives of the electric field and
time derivatives of the magnetic field. In the phasor
version of the MFE, the time derivative operator has
been replaced with multiplication by jω. This is a
tremendous simplification since the equations now
involve differentiation over position only.
Furthermore, no information is lost in this
simplification – for a reminder of why that is, see the
discussion of Fourier Analysis at the end of
Section 1.5. Without this kind of simplification, much
of what is now considered “basic” engineering
electromagnetics would be intractable.
The procedure for conversion of the remaining two
equations is very similar, yielding:
e =0
∇·B
e =J
e + jω D
e
∇×H
The details are left as an exercise for the reader.
(9.18)
(9.19)
195
The differential form of Maxwell’s Equations
(Equations 9.10, 9.17, 9.18, and 9.19) involve
operations on the phasor representations of the
physical quantities. These equations have the advantage that differentiation with respect to time
is replaced by multiplication by jω.
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
196
9.2 Wave Equations for
Source-Free and Lossless
Regions
[m0036]
Electromagnetic waves are solutions to a set of
coupled differential simultaneous equations – namely,
Maxwell’s Equations. The general solution to these
equations includes constants whose values are
determined by the applicable electromagnetic
boundary conditions. However, this direct approach
can be difficult and is often not necessary. In
unbounded homogeneous regions that are “source
free” (containing no charges or currents), a simpler
approach is possible. In this section, we reduce
Maxwell’s Equations to wave equations that apply to
the electric and magnetic fields in this simpler
category of scenarios. Before reading further, the
reader should consider a review of Section 1.3 (noting
in particular Equation 1.1) and Section 3.6 (wave
equations for voltage and current on a transmission
line). This section seeks to develop the analogous
equations for electric and magnetic waves.
We can get the job done using the differential “point”
phasor form of Maxwell’s Equations, developed in
Section 9.1. Here they are:
e = ρev
∇·D
(9.20)
e = −jω B
e
∇×E
(9.21)
e =J
e + jω D
e
∇×H
(9.23)
e =0
∇·D
(9.24)
e =0
∇·B
(9.22)
In a source-free region, there is no net charge and no
e = 0 in the present
current, hence ρev = 0 and J
analysis. The above equations become
e = −jω B
e
∇×E
(9.25)
e = +jω D
e
∇×H
(9.27)
e =0
∇·B
e = ǫE
e and that ǫ is a
Next, we recall that D
real-valued constant for a medium that is
(9.26)
homogeneous, isotropic, and linear (Section 2.8).
e = µH̃ and µ is a real-valued constant.
Similarly, B
Thus, under these conditions, it is sufficient to
e or E
e and either B
e or H.
e The
consider either D
choice is arbitrary, but in engineering applications it is
e and H.
e Eliminating the
customary to use E
e and B,
e the above
now-redundant quantities D
equations become
e =0
∇·E
e = −jωµH
e
∇×E
e =0
∇·H
e = +jωǫE
e
∇×H
(9.28)
(9.29)
(9.30)
(9.31)
It is important to note that requiring the region of
interest to be source-free precludes the possibility of
loss in the medium. To see this, let’s first be clear
about what we mean by “loss.” For an
electromagnetic wave, loss is observed as a reduction
in the magnitude of the electric and magnetic field
with increasing distance. This reduction is due to the
dissipation of power in the medium. This occurs
when the conductivity σ is greater than zero because
e = σ E;
e
Ohm’s Law for Electromagnetics (J
Section 6.3) requires that power in the electric field be
transferred into conduction current, and is thereby lost
to the wave (Section 6.6). When we required J to be
zero above, we precluded this possibility; that is, we
implicitly specified σ = 0. The fact that the
constitutive parameters µ and ǫ appear in
Equations 9.28–9.31, but σ does not, is further
evidence of this.
Equations 9.28–9.31 are Maxwell’s Equations for
a region comprised of isotropic, homogeneous,
and source-free material. Because there can be no
conduction current in a source-free region, these
equations apply only to material that is lossless
(i.e., having negligible σ).
Before moving on, one additional disclosure is
appropriate. It turns out that there actually is a way to
use Equations 9.28–9.31 for regions in which loss is
significant. This requires a redefinition of ǫ as a
complex-valued quantity. We shall not consider this
technique in this section. We mention this simply
because one should be aware that if permittivity
9.2. WAVE EQUATIONS FOR SOURCE-FREE AND LOSSLESS REGIONS
appears as a complex-valued quantity, then the
imaginary part represents loss.
To derive the wave equations we begin with the MFE,
Equation 9.29. Taking the curl of both sides of the
equation we obtain
e = ∇ × −jωµH
e
∇× ∇×E
e
= −jωµ ∇ × H
(9.32)
e using
On the right we can eliminate ∇ × H
Equation 9.31:
e = −jωµ +jωǫE
e
−jωµ ∇ × H
e
= +ω 2 µǫE
(9.33)
On the left side of Equation 9.32, we apply the vector
identity
2
∇ × ∇ × A = ∇ (∇ · A) − ∇ A
(9.34)
which in this case is
e =∇ ∇·E
e − ∇2 E
e
∇×∇×E
197
frequency, permittivity, and permeability all in one
constant. Another reason is to emphasize the
connection to the parameter β appearing in
transmission line theory (see Section 3.8 for a
reminder). It should be clear that β is a phase
propagation constant, having units of 1/m (or rad/m,
if you prefer), and indicates the rate at which the
phase of the propagating wave progresses with
distance.
e is obtained using essentially
The wave equation for H
the same procedure, which is left as an exercise for
the reader. It should be clear from the duality
apparent in Equations 9.28-9.31 that the result will be
very similar. One finds:
e + β2H
e =0
∇2 H
(9.40)
Equations 9.39 and 9.40 are the wave equations
e and H,
e respectively, for a region comprised
for E
of isotropic, homogeneous, lossless, and sourcefree material.
(9.35)
We may eliminate the first term on the right using
Equation 9.28, yielding
e = −∇2 E
e
∇×∇×E
(9.36)
e + ω 2 µǫE
e =0
∇2 E
(9.37)
Substituting these results back into Equation 9.32 and
rearranging terms we have
e Note that it is a
This is the wave equation for E.
homogeneous (in the mathematical sense of the word)
differential equation, which is expected since we have
derived it for a source-free region.
e and H
e are solutions to the
Looking ahead, note that E
same homogeneous differential equation.
e and H
e cannot be different by more
Consequently, E
than a constant factor and a direction. In fact, we can
also determine something about the factor simply by
e has units of
examining the units involved: Since E
e
V/m and H has units of A/m, this factor will be
expressible in units of the ratio of V/m to A/m, which
is Ω. This indicates that the factor will be an
impedance. This factor is known as the wave
impedance and will be addressed in Section 9.5. This
impedance is analogous the characteristic impedance
of a transmission line (Section 3.7).
Additional Reading:
It is common to make the following definition
√
β , ω µǫ
(9.38)
• “Electromagnetic Wave Equation” on Wikipedia.
so that Equation 9.37 may be written
e + β2E
e =0
∇2 E
• “Wave Equation” on Wikipedia.
(9.39)
Why go the the trouble of defining β? One reason is
that β conveniently captures the contribution of the
198
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
9.3 Types of Waves
[m0142]
Solutions to the electromagnetic wave equations
(Section 9.2) exist in a variety of forms, representing
different types of waves. It is useful to identify three
particular geometries for unguided waves. Each of
these geometries is defined by the shape formed by
surfaces of constant phase, which we refer to as
phasefronts. (Keep in mind the analogy between
electromagnetic waves and sound waves (described in
Section 1.3), and note that sound waves also exhibit
these geometries.)
A spherical wave has phasefronts that form
concentric spheres, as shown in Figure 9.1. Waves are
well-modeled as spherical when the dimensions of the
source of the wave are small relative to the scale at
which the wave is observed. For example, the wave
radiated by an antenna having dimensions of 10 cm,
when observed in free space over a scale of 10 km,
appears to have phasefronts that are very nearly
spherical. Note that the magnitude of the field on a
phasefront of a spherical wave may vary significantly,
but it is the shape of phasefronts that make it a
spherical wave.
c Y. Qin CC BY 4.0
Figure 9.1: The phasefronts of a spherical wave form
concentric spheres.
A cylindrical wave exhibits phasefronts that form
concentric cylinders, as shown in Figure 9.2. Said
differently, the phasefronts of a cylindrical wave are
circular in one dimension, and planar in the
perpendicular direction. A cylindrical wave is often a
good description of the wave that emerges from a
line-shaped source.
A plane wave exhibits phasefronts that are planar,
with planes that are parallel to each other as shown in
Figure 9.3. There are two conditions in which waves
are well-modeled as plane waves. First, some
structures give rise to waves that appear to have
planar phasefronts over a limited area; a good
example is the wave radiated by a parabolic reflector,
as shown in Figure 9.4. Second, all waves are
well-modeled as plane waves when observed over a
small region located sufficiently far from the source.
In particular, spherical waves are “locally planar” in
the sense that they are well-modeled as planar when
observed over a small portion of the spherical
c Y. Qin CC BY 4.0
Figure 9.2: The phasefronts of a cylindrical wave
form concentric cylinders.
9.4. UNIFORM PLANE WAVES: DERIVATION
199
"locally-planar" phasefronts
spherical phasefront
c Y. Qin CC BY 4.0
Figure 9.3: The phasefronts of a plane wave form
parallel planes.
c Y. Qin CC BY 4.0
reflector
Figure 9.5:
“Locally planar” approximation of a
spherical wave over a limited area.
feed
9.4 Uniform Plane Waves:
Derivation
[m0038]
planar ponts
c Y. Qin CC BY 4.0
Figure 9.4: Plane waves formed in the region in front
of a parabolic reflector antenna.
phasefront, as shown in Figure 9.5. An analogy is that
the Earth seems “locally flat” to an observer on the
ground, even though it is clearly spherical to an
observer in orbit. The “locally planar” approximation
is often employed because it is broadly applicable and
simplifies analysis.
Most waves are well-modeled as spherical, cylindrical, or plane waves.
Plane waves (having planar phasefronts) are of
particular importance due to wide applicability of
the “locally planar” approximation.
Section 9.2 showed how Maxwell’s Equations could
be reduced to a pair of phasor-domain “wave
equations,” namely:
e + β2E
e =0
∇2 E
(9.41)
e + β2H
e =0
∇2 H
(9.42)
√
where β = ω µǫ, assuming unbounded
homogeneous, isotropic, lossless, and source-free
media. In this section, we solve these equations for
the special case of a uniform plane wave. A uniform
e and H
e have
plane wave is one for which both E
constant magnitude and phase in a specified plane.
Despite being a special case, the solution turns out to
be broadly applicable, appearing as a common
building block in many practical and theoretical
problems in unguided propagation (as explained in
Section 9.3), as well as in more than a few
transmission line and waveguide problems.
e
To begin, let us assume that the plane over which E
e
and H have constant magnitude and phase is a plane
of constant z. First, note that we may make this
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
200
Decomposing this equation into separate equations
assumption with no loss of generality. For example,
for each of the three coordinates, we obtain the
we could alternatively select a plane of constant y,
following:
solve the problem, and then simply exchange
variables to get a solution for planes of constant z (or
2
∂
∂2
∂2 e
x).1 Furthermore, the solution for any planar
ex = 0
(9.51)
+ 2 + 2 Ex + β 2 E
∂x2
∂y
∂z
orientation not corresponding to a plane of constant x,
y, or z may be similarly obtained by a rotation of
2
∂2
∂2 e
∂
coordinates, since the physics of the problem does not
ey = 0
(9.52)
+ 2 + 2 Ey + β 2 E
∂x2
∂y
∂z
depend on the orientation of this plane – if it does,
2
then the medium is not isotropic!
∂2
∂2 e
∂
ez = 0
(9.53)
+ 2 + 2 Ez + β 2 E
∂x2
∂y
∂z
We may express the constraint that the magnitude and
e and H
e are constant over a plane that is
phase of E
Applying the constraints of Equations 9.46 and 9.47,
perpendicular to the z axis as follows:
we note that many of the terms in
∂ e
∂ e
∂ e
∂ e
E=
E=
H=
H=0
∂x
∂y
∂x
∂y
Equations 9.51–9.53 are zero. We are left with:
(9.43)
Let us identify the Cartesian components of each of
these fields as follows:
e = x̂E
ex + ŷE
ey + ẑE
ez , and
E
e = x̂H
e x + ŷH
e y + ẑH
ez
H
(9.44)
(9.45)
e
Now Equation 9.43 may be interpreted in detail for E
as follows:
∂ e
∂ e
∂ e
Ex =
Ey =
Ez = 0
∂x
∂x
∂x
∂ e
∂ e
∂ e
Ex =
Ey =
Ez = 0
∂y
∂y
∂y
e as follows:
and for H
∂ e
∂ e
∂ e
Hx =
Hy =
Hz = 0
∂x
∂x
∂x
∂ e
∂ e
∂ e
Hx =
Hy =
Hz = 0
∂y
∂y
∂y
(9.46)
(9.47)
(9.48)
(9.49)
e (Equation 9.41) written
The wave equation for E
explicitly in Cartesian coordinates is
∂2
∂2
∂2 e
ey + ẑE
ez
+
+
x̂
E
+
ŷ
E
x
∂x2
∂y 2
∂z 2
ex + ŷE
ey + ẑE
ez = 0
+β 2 x̂E
(9.50)
1 By the way, this is a highly-recommended exercise for the student.
∂2 e
ex = 0
Ex + β 2 E
∂z 2
∂2 e
ey = 0
Ey + β 2 E
∂z 2
∂2 e
ez = 0
Ez + β 2 E
∂z 2
(9.54)
(9.55)
(9.56)
Now we will show that Equation 9.43 also implies
ez must be zero. To show this, we use Ampere’s
that E
Law for a source-free region (Section 9.2):
e = +jωǫE
e
∇×H
and take the dot product with ẑ on both sides:
e = +jωǫE
ez
ẑ · ∇ × H
∂ e
∂ e
ez
Hx −
Hy = +jωǫE
∂y
∂x
(9.57)
(9.58)
Again applying the constraints of Equation 9.43, the
left side of Equation 9.58 must be zero; therefore,
e
ez = 0. The exact same procedure applied to H
E
(using the Maxwell-Faraday Equation; also given in
e z is also zero.2 Here is
Section 9.2) reveals that H
what we have found:
If a wave is uniform over a plane, then the electric
and magnetic field vectors must lie in this plane.
This conclusion is a direct consequence of the fact
that Maxwell’s Equations require the electric field to
2 Showing this is a highly-recommended exercise for the reader.
9.4. UNIFORM PLANE WAVES: DERIVATION
201
be proportional to the curl of the magnetic field and
vice-versa.
will not provide a rigorous proof of this, one way to
see that this is true is to imagine that any type of wave
can be interpreted as the sum (formally, a linear
combination) of uniform plane waves, so
perpendicular orientation of the field vectors with
respect to the direction of propagation is inescapable.
The general solution to Equation 9.54 is:
ex = E + e−jβz + E − e+jβz
E
x0
x0
(9.59)
ey = E + e−jβz + E − e+jβz
E
y0
y0
(9.60)
e = x̂E
ex + ŷE
ey
E
(9.61)
−
+
are complex-valued constants.
and Ex0
where Ex0
The same procedure yields the uniform plane wave
The values of these constants are determined by
e which is
solution to the wave equation for H,
boundary conditions – possibly sources – outside the
e = x̂H
e x + ŷH
ey
region of interest. Since in this section we are limiting
H
(9.62)
our scope to source-free and homogeneous regions,
where
+
we may for the moment consider the values of Ex0
−
e x = H + e−jβz + H − e+jβz
and Ex0 to be arbitrary, since any values will satisfy
H
(9.63)
x0
x0
the associated wave equation.3
e y = H + e−jβz + H − e+jβz
H
(9.64)
y0
y0
e
Similarly we have for Ey :
+
−
+
−
and where H , H , H and H are arbitrary
x0
+
−
where Ey0
and Ey0
are again arbitrary constants.
Summarizing, we have found
ex and E
ey are given by Equations 9.59 and
where E
9.60, respectively.
Note that Equations 9.59 and 9.60 are essentially the
same equations encountered in the study of waves in
lossless transmission lines; for a reminder, see
Section 3.6. Specifically, factors containing e−jβz
describe propagation in the +z direction, whereas
factors containing e+jβz describe propagation in the
−z direction. We conclude:
If a wave is uniform over a plane, then possible
directions of propagation are the two directions
perpendicular to the plane.
Since we previously established that the electric and
magnetic field vectors must lie in the plane, we also
conclude:
The direction of propagation is perpendicular to
the electric and magnetic field vectors.
This conclusion turns out to be generally true; i.e., it
is not limited to uniform plane waves. Although we
3 The reader is encouraged to confirm that these are solutions by
substitution into the associated wave equation.
x0
y0
y0
constants. Note that the solution is essentially the
e with the sole difference being that
same as that for E,
the arbitrary constants may apparently have different
values.
To this point, we have seen no particular relationship
between the electric and magnetic fields, and it may
appear that the electric and magnetic fields are
independent of each other. However, Maxwell’s
Equations – specifically, the two curl equations –
make it clear that there must be a relationship
between these fields. Subsequently the arbitrary
e and H
e must also be
constants in the solutions for E
related. In fact, there are two considerations here:
e must be related to
• The magnitude and phase of E
e
the magnitude and phase of H. Since both fields
are solutions to the same differential (wave)
equation, they may differ by no more than a
e and
multiplicative constant. Since the units of E
e
H are V/m and A/m respectively, this constant
must be expressible in units of V/m divided by
A/m; i.e., in units of Ω, an impedance.
e must be related to direction
• The direction of E
e
of H.
Let us now address these considerations. Consider an
electric field that points in one of the cardinal
directions – let’s say +x̂ – and make the definition
+
E0 , Ex0
for notational convenience. Then the
electric field intensity may be written as follows:
e = x̂E0 e−jβz
E
(9.65)
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
202
Again, there is no loss of generality here since the
coordinate system could be rotated in such a way that
any uniform plane could be described in this way.
e from E
e using the
We may now determine H
Maxwell-Faraday Equation (Section 9.2):
e = −jωµH
e
∇×E
e we find:
Solving this equation for H,
e
∇ × x̂E0 e−jβz
∇×E
e
H=
=
−jωµ
−jωµ
E
(9.66)
ez = 0, so we have simply
Here E
e
e = ∂ Ex
ŷ · ∇ × E
∂z
(9.67)
(9.68)
(9.69)
(9.70)
ez ,
It is not necessary to repeat this procedure for H
e must be
since we know in advance that H
perpendicular to the direction of propagation and
e z = 0. Returning to Equation 9.67,
subsequently H
we obtain:
1 ∂ e
Ex
−jωµ ∂z
1 ∂
E0 e−jβz
= ŷ
−jωµ ∂z
1
= ŷ
−jβE0 e−jβz
−jωµ
β
= ŷ
E0 e−jβz
ωµ
no
fp
ey = E
ez = 0, the above expression is zero and
Since E
e x = 0. Next, the ŷ component:
subsequently H
e
e
e = ∂ Ex − ∂ E z
ŷ · ∇ × E
∂z
∂x
dir
ec
tio
Now let us apply the curl operator. The complete
expression for the curl operator in Cartesian
coordinates is given in Section B.2. Here let us
consider one component at a time, starting with the x̂
component:
e
e
e = ∂ Ez − ∂ Ey
x̂ · ∇ × E
∂y
∂z
H
e = ŷ
H
ro
!"
Figure 9.6: Relationship between the electric field direction, magnetic field direction, and direction of propagation.
direction of propagation. However, we have now
e and H
e are
found a more specific relationship: E
perpendicular to each other. Just as x̂ × ŷ = ẑ, we
e ×H
e points in the direction of propagation.
see that E
This is illustrated in Figure 9.6. Summarizing:
e H,
e and the direction of propagation E×
e H
e are
E,
mutually perpendicular.
Now let us resolve the question of the factor relating
e and H.
e The factor is now seen to be β/ωµ in
E
Equation 9.71, which can be simplified as follows:
√
ω µǫ
β
1
(9.72)
=
=p
ωµ
ωµ
µ/ǫ
p
The factor µ/ǫ appearing above has units of Ω is
known variously as the wave impedance or the
intrinsic impedance of the medium. Assigning this
quantity the symbol “η,” we have:
ex
E
η,
=
e
Hy
(9.71)
e points in the +ŷ direction. So, as
Note that H
e and H
e are perpendicular to the
expected, both E
E x
r
µ
ǫ
(9.73)
The ratio of the electric field intensity to the
magnetic field intensity is the wave impedance η
(Equation 9.73; units of Ω). In lossless media, η
is determined by the ratio of permeability of the
medium to the permittivity of the medium.
9.5. UNIFORM PLANE WAVES: CHARACTERISTICS
The wave impedance in free space, assigned the
symbol η0 , is
r
µ0 ∼
η0 ,
(9.74)
= 377 Ω.
ǫ0
e is as
Wrapping up our solution, we find that if E
given by Equation 9.65, then
e = ŷ E0 e−jβz
H
η
(9.75)
203
9.5 Uniform Plane Waves:
Characteristics
[m0039]
In Section 9.4, expressions for the electric and
magnetic fields are determined for a uniform plane
wave in lossless media. If the planar phasefront is
perpendicular to the z axis, then waves may propagate
in either the +ẑ direction of the −ẑ direction. If we
e to point in the
consider only the former, and select E
+x̂ direction, then we find
e = +x̂E0 e−jβz
E
(9.76)
e = +ŷ E0 e−jβz
H
(9.77)
η
√
where β = ω µǫ is the phase propagation constant,
p
η = µ/ǫ is the wave impedance, and E0 is a
complex-valued constant associated with the
magnitude and phase of the source. This result is in
fact completely general for uniform plane waves,
since any other possibility may be obtained by simply
rotating coordinates. In fact, this is pretty easy
e H,
e and the
because (as determined in Section 9.4) E,
direction of propagation are mutually perpendicular,
with the direction of propagation pointing in the same
e × H.
e
direction as E
In this section, we identify some important
characteristics of uniform plane waves, including
wavelength and phase velocity. Chances are that
much of what appears here will be familiar to the
reader; if not, a quick review of Sections 1.3
(“Fundamentals of Waves”) and 3.8 (“Wave
Propagation on a Transmission Line”) are
recommended.
e and H
e are phasors representing
First, recall that E
physical (real-valued) fields and are not the field
values themselves. The actual, physical electric field
intensity is
o
n
e jωt
E = Re Ee
(9.78)
= Re x̂E0 e−jβz ejωt
= x̂ |E0 | cos (ωt − βz + ψ)
where ψ is the phase of E0 . Similarly:
H = ŷ
|E0 |
cos (ωt − βz + ψ)
η
(9.79)
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
204
x
√
substitution β = ω µǫ, we find
E
z
y
H
c E. Boutet (Modified) CC BY SA 3.0
Figure 9.7: Relationship between the electric field direction, magnetic field direction, and direction of propagation (here, +ẑ).
This result is illustrated in Figure 9.7. Note that both
E and H (as well as their phasor representations)
have the same phase and have the same frequency and
position dependence cos (ωt − βz + ψ). Since β is a
real-valued constant for lossless media, only the
phase, and not the magnitude, varies with z. If ω → 0,
then β → 0 and the fields no longer depend on z; in
this case, the field is not propagating. For ω > 0,
cos (ωt − βz + ψ) is periodic in z; specifically, it has
the same value each time z increases or decreases by
2π/β. This is, by definition, the wavelength λ:
2π
β
vp =
ω
1
√ =√
ω µǫ
µǫ
(9.82)
Note that vp , like the wave impedance η, depends
only on material properties. For example, the phase
velocity of an electromagnetic wave in free space,
given the special symbol c, is
1
∼
= 3.00 × 108 m/s
µ0 ǫ0
(9.83)
This constant is commonly referred to as the speed of
light, but in fact it is the phase velocity of an
electromagnetic field at any frequency (not just
optical frequencies) in free space. Since the
permittivity ǫ and permeability µ of any material is
greater than that of a vacuum, vp in any material is
less than the phase velocity in free space.
Summarizing:
c , vp |µ=µ0 ,ǫ=ǫ0 = √
Phase velocity is the speed at which any point of
constant phase appears to travel along the direction of propagation.
(9.80)
Phase velocity is maximum (= c) in free space,
√
and slower by a factor of 1/ µr ǫr in any other
lossless medium.
If we observe this wave at some fixed point (i.e., hold
z constant), we find that the electric and magnetic
fields are also periodic in time; specifically, they have
the same value each time t increases by 2π/ω. We
may characterize the speed at which the wave travels
by comparing the distance required to experience 2π
of phase rotation at a fixed time, which is 1/β; to the
time it takes to experience 2π of phase rotation at a
fixed location, which is 1/ω. This is known as the
phase velocity4 vp :
Finally, we note the following relationship between
frequency f , wavelength, and phase velocity:
λ,
vp ,
1/β
ω
=
1/ω
β
(9.81)
λ=
2π
2π
vp
2π
=
=
=
β
ω/vp
2πf /vp
f
(9.84)
Thus, given any two of the parameters f , λ, and vp ,
we may solve for the remaining quantity. Also note
that as a consequence of the inverse-proportional
relationship between λ and vp , we find:
At a given frequency, the wavelength in any material is shorter than the wavelength in free space.
Note that vp has the units expected from its definition,
namely (rad/s)/(rad/m) = m/s. If we make the
Example 9.1. Wave propagation in a lossless
dielectric.
4 We acknowledge that this is a misnomer, since velocity is properly defined as speed in a specified direction, and vp by itself does
not specify direction. In fact, the phase velocity in this case is properly said to be +ẑvp . Nevertheless, we adopt the prevailing terminology.
Polyethylene is a low-loss dielectric having
ǫr ∼
= 2.3. What is the phase velocity in
polyethylene? What is wavelength in
9.5. UNIFORM PLANE WAVES: CHARACTERISTICS
polyethylene? The frequency of interest is
1 GHz.
Solution. Low-loss dielectrics exhibit µr ∼
=1
and σ ≈ 0. Therefore the phase velocity is
c
1
=√ ∼
= 1.98 × 108 m/s
µ0 ǫr ǫ0
ǫr
(9.85)
i.e., very nearly two-thirds of the speed of light
in free space. The wavelength at 1 GHz is
vp = √
λ=
vp ∼
= 19.8 cm
f
(9.86)
Again, about two-thirds of a wavelength at the
same frequency in free space.
Returning to polarization and magnitude, it is useful
to note that Equation 9.79 could be written in terms of
E as follows:
1
(9.87)
H = ẑ × E
η
i.e., H is perpendicular to both the direction of
propagation (here, +ẑ) and E, with magnitude that is
different from E by the wave impedance η. This
simple expression is very useful for quickly
determining H given E when the direction of
propagation is known. In fact, it is so useful that it
commonly explicitly generalized as follows:
H=
1
k̂ × E
η
(9.88)
where k̂ (here, +ẑ) is the direction of propagation.
Similarly, given H, k̂, and η, we may find E using
E = −η k̂ × H
(9.89)
These spatial relationships can be readily verified
using Figure 9.7.
Equations 9.88 and 9.89 are known as the plane
wave relationships.
The plane wave relationships apply equally well to
the phasor representations of E and H; i.e.,
e = −η k̂ × H
e
E
e
e = 1 k̂ × E
H
η
(9.90)
(9.91)
205
These equations can be readily verified by applying
the definition of the associated phasors (e.g.,
Equation 9.78). It also turns out that these
relationships apply at each point in space, even if the
waves are not planar, uniform, or in lossless media.
Example 9.2. Analysis of a radially-directed
plane wave.
Consider the scenario illustrated in Figure 9.8.
Here a uniform plane wave having frequency
f = 3 GHz is propagating along a path of
constant φ, where is φ is known but not
specified. The phase of the electric field is
π/2 radians at ρ = 0 (the origin) and t = 0. The
material is an effectively unbounded region of
free space. The electric field is oriented in the
+ẑ direction and has peak magnitude of
1 mV/m. Find (a) the electric field intensity in
phasor representation, (b) the magnetic field
intensity in phasor representation, and (c) the
actual, physical electric field along the radial
path.
Solution: First, realize that this
“radially-directed” plane wave is in fact a plane
wave, and not a cylindrical wave. It may well we
be that if we “zoom out” far enough, we are able
to perceive a cylindrical wave (for more on this
idea, see Section 9.3), or it might simply be that
wave is exactly planar, and cylindrical
coordinates just happen to be a convenient
coordinate system for this application. In either
case, the direction of propagation k̂ = +ρ̂ and
the solution to this example will be the same.
Here’s the phasor representation for the electric
field intensity of a uniform plane wave in a
lossless medium, analogous to Equation 9.76:
e = +ẑE0 e−jβρ
E
(9.92)
e = +ẑj |E0 | e−jβρ
E
(9.93)
From the problem statement, |E0 | = 1 mV/m.
Also from the problem statement, the phase of
E0 is π/2 radians; in fact, we could just write
E0 = +j |E0 |. Thus, the answer to (a) is
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
206
y
where the propagation constant
√
√
β = ω µǫ = 2πf µ0 ǫ0 ∼
= 62.9 rad/m.
The answer to (b) is easiest to obtain from the
plane wave relationship:
E
e = 1 k̂ × E
H
η
1
= ρ̂ × +ẑj |E0 | e−jβρ
η
j |E0 | −jβρ
e
(9.94)
= −φ̂
η
p
where η here is µ0 /ǫ0 ∼
= 377 Ω. Thus, the
answer to part (b) is
e = −φ̂jH0 e−jβρ
H
(9.95)
∼ 2.65 µA/m. At this point you
where H0 =
e ×H
e should
should check vector directions: E
point in the direction
of
propagation
+ρ̂.
Here
we find ẑ × −φ̂ = +ρ̂, as expected.
The answer to (c) is obtained by applying the
defining relationship for phasors to the answer to
part (a):
n
o
e jωt
E = Re Ee
= Re +ẑj |E0 | e−jβρ ejωt
n
o
= +ẑ |E0 | Re ejπ/2 e−jβρ ejωt
π
(9.96)
= +ẑ |E0 | cos ωt − βρ +
2
Additional Reading:
• “Electromagnetic radiation” on Wikipedia.
ϕ
x
Figure 9.8: A radially-directed plane wave.
9.6 Wave Polarization
[m0131]
Polarization refers to the orientation of the electric
field vector. For waves, the term “polarization” refers
specifically to the orientation of this vector with
increasing distance along the direction of propagation,
or, equivalently, the orientation of this vector with
increasing time at a fixed point in space. The relevant
concepts are readily demonstrated for uniform plane
waves, as shown in this section. A review of
Section 9.5 (“Uniform Plane Waves: Characteristics”)
is recommended before reading further.
To begin, consider the following uniform plane wave,
described in terms of the phasor representation of its
electric field intensity:
e x = x̂Ex e−jβz
E
(9.97)
e y = ŷEy e−jβz
E
(9.98)
Here Ex is a complex-valued constant representing
the magnitude and phase of the wave, and β is the
positive real-valued propagation constant. Therefore,
this wave is propagating in the +ẑ direction in
lossless media. This wave is said to exhibit linear
polarization (and “linearly polarized”) because the
electric field always points in the same direction,
namely +x̂. Now consider the wave
e y is identical to E
e x except that the electric
Note that E
field vector now points in the +ŷ direction and has
magnitude and phase that is different by the factor
9.6. WAVE POLARIZATION
207
Linear polarization arises when the source of the wave
is linearly polarized. A common example is the wave
radiated by a straight wire antenna, such as a dipole
or a monopole. Linear polarization may also be
created by passing a plane wave through a polarizer;
this is particularly common at optical frequencies (see
“Additional Reading” at the end of this section).
x
z
A commonly-encountered alternative to linear
polarization is circular polarization. For an
explanation, let us return to the linearly-polarized
e x and E
e y defined earlier. If both of
plane waves E
these waves exist simultaneously, then the total
electric field intensity is simply the sum:
e =E
ex + E
ey
E
by RJB1 (Modified)
= (x̂Ex + ŷEy ) e−jβz
(9.102)
Figure 9.9: Linear polarization. Here Ex is shown in
blue, Ey is shown in green, and E is shown in red with
black vector symbols. In this example the phases of If the phase of Ex and Ey is the same, then
Ex = Eρ cos φ, Ey = Eρ sin φ, and the above
Ex and Ey are zero and φ = −π/4.
expression is essentially the same as Equation 9.101.
e is linearly polarized. But what if the
In this case, E
phases
of
E
Ey /Ex . This wave too is said to exhibit linear
x and Ey are different? In particular, let’s
consider
the
following case. Let Ex = E0 , some
polarization, because, again, the direction of the
complex-valued
constant, and let Ey = +jE0 , which
electric field is constant with both time and position.
is
E
phase-shifted
by +π/2 radians. With no further
In fact, all linearly-polarized uniform plane waves
0
e x and E
e y are different only
propagating in the +ẑ direction in lossless media can math, it is apparent that E
in that one is phase-shifted by π/2 radians relative to
be described as follows:
the other. For the physical (real-valued) fields, this
e = ρ̂Eρ e−jβz
E
(9.99) means that Ex has maximum magnitude when Ey is
zero and vice versa. As a result, the direction of
This is so because ρ̂ could be x̂, ŷ, or any other
E = Ex + Ey will rotate in the x − y plane, as shown
direction that is perpendicular to ẑ. If one is
in Figure 9.10
determined to use Cartesian coordinates, the above
expression may be rewritten using (Section 4.3)
The rotation of the electric field vector can also be
identified mathematically. When Ex = E0 and
ρ̂ = x̂ cos φ + ŷ cos φ
(9.100) Ey = +jE0 , Equation 9.102 can be written:
yielding
e = (x̂ cos φ + ŷ cos φ) Eρ e−jβz
E
(9.101)
When written in this form, φ = 0 corresponds to
e =E
e x , φ = π/2 corresponds to E
e =E
e y , and any
E
other value of φ corresponds to some other constant
orientation of the electric field vector; see Figure 9.9
for an example.
A wave is said to exhibit linear polarization if the
direction of the electric field vector does not vary
with either time or position.
e = (x̂ + j ŷ) E0 e−jβz
E
(9.103)
Now reverting from phasor notation to the physical
field:
n
o
e jωt
E = Re Ee
= Re (x̂ + j ŷ) E0 e−jβz ejωt
π
= x̂ |E0 | cos (ωt − βz) + ŷ |E0 | cos ωt − βz +
2
(9.104)
As anticipated, we see that both Ex and Ey vary
sinusoidally, but are π/2 radians out of phase
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
208
by Dave3457
Figure 9.10: A circularly-polarized wave (in red, with
black vector symbols) resulting from the addition of
orthogonal linearly-polarized waves (shown in green
and blue) that are phase-shifted by π/2 radians.
resulting in rotation in the plane perpendicular to the
direction of propagation.
In the example above, the electric field vector rotates
either clockwise or counter-clockwise relative to the
direction of propagation. The direction of this
rotation can be identified by pointing the thumb of the
left hand in the direction of propagation; in this case,
the fingers of the left hand curl in the direction of
rotation. For this reason, this particular form of
circular polarization is known as left circular (or
”left-hand” circular) polarization (LCP). If we instead
had chosen Ey = −jE0 = −jEx , then the direction
of E rotates in the opposite direction, giving rise to
right circular (or “right-hand” circular) polarization
(RCP). These two conditions are illustrated in
Figure 9.11.
A wave is said to exhibit circular polarization
if the electric field vector rotates with constant
magnitude. Left- and right-circular polarizations
may be identified by the direction of rotation with
respect to the direction of propagation.
In engineering applications, circular polarization is
useful when the relative orientations of transmit and
receive equipment is variable and/or when the
medium is able rotate the electric field vector. For
example, radio communications involving satellites in
by Dave3457
Figure 9.11: Left-circular polarization (LCP; top) and
right-circular polarization (RCP; bottom).
9.7. WAVE POWER IN A LOSSLESS MEDIUM
non-geosynchronous orbits typically employ circular
polarization. In particular, satellites of the U.S.
Global Positioning System (GPS) transmit circular
polarization because of the variable geometry of the
space-to-earth radio link and the tendency of the
Earth’s ionosphere to rotate the electric field vector
through a mechanism known Faraday rotation
(sometimes called the “Faraday effect”). If GPS were
instead to transmit using a linear polarization, then a
receiver would need to continuously adjust the
orientation of its antenna in order to optimally receive
the signal. Circularly-polarized radio waves can be
generated (or received) using pairs of
perpendicularly-oriented dipoles that are fed the same
signal but with a 90◦ phase shift, or alternatively by
using an antenna that is intrinsically
circularly-polarized, such as a helical antenna (see
“Additional Reading” at the end of this section).
Linear and circular polarization are certainly not the
only possibilities. Elliptical polarization results when
Ex and Ey do not have equal magnitude. Elliptical
polarization is typically not an intended condition, but
rather is most commonly observed as a degradation in
a system that is nominally linearly- or
circularly-polarized. For example, most antennas that
are said to be “circularly polarized” instead produce
circular polarization only in one direction and various
degrees of elliptical polarization in all other
directions.
Additional Reading:
• “Polarization (waves)” on Wikipedia.
• “Dipole antenna” on Wikipedia.
• “Polarizer” on Wikipedia.
• “Faraday effect” on Wikipedia.
• “Helical antenna” on Wikipedia.
209
9.7 Wave Power in a Lossless
Medium
[m0041]
In many applications involving electromagnetic
waves, one is less concerned with the instantaneous
values of the electric and magnetic fields than the
power associated with the wave. In this section, we
address the issue of how much power is conveyed by
an electromagnetic wave in a lossless medium. The
relevant concepts are readily demonstrated in the
context of uniform plane waves, as shown in this
section. A review of Section 9.5 (“Uniform Plane
Waves: Characteristics”) is recommended before
reading further.
Consider the following uniform plane wave,
described in terms of the phasor representation of its
electric field intensity:
e = x̂E0 e−jβz
E
(9.105)
Here E0 is a complex-valued constant associated with
the source of the wave, and β is the positive
real-valued propagation constant. Therefore, the wave
is propagating in the +ẑ direction in lossless media.
The first thing that should be apparent is that the
amount of power conveyed by this wave is infinite.
The reason is as follows. If the power passing through
any finite area is greater than zero, then the total
power must be infinite because, for a uniform plane
wave, the electric and magnetic field intensities are
constant over a plane of infinite area. In practice, we
never encounter this situation because all practical
plane waves are only “locally planar” (see Section 9.3
for a refresher on this idea). Nevertheless, we seek
some way to express the power associated with such
waves.
The solution is not to seek total power, but rather
power per unit area. This quantity is known as the
spatial power density, or simply “power density,” and
has units of W/m2 .5 Then, if we are interested in total
power passing through some finite area, then we may
5 Be careful: The quantities power spectral density (W/Hz) and
power flux density (W/(m2 ·Hz)) are also sometimes referred to as
“power density.” In this section, we will limit the scope to spatial
power density (W/m2 ).
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
210
simply integrate the power density over this area.
Let’s skip to the answer, and then consider where this
answer comes from. It turns out that the
instantaneous power density of a uniform plane wave
is the magnitude of the Poynting vector
S,E×H
(9.106)
Note that this equation is dimensionally correct; i.e.
the units of E (V/m) times the units of H (A/m) yield
the units of spatial power density (V·A/m2 , which is
W/m2 ). Also, the direction of E × H is in the
direction of propagation (reminder: Section 9.5),
which is the direction in which the power is expected
to flow. Thus, we have some compelling evidence that
|S| is the power density we seek. However, this is not
proof – for that, we require the Poynting Theorem,
which is a bit outside the scope of the present section,
but is addressed in the “Additional Reading” at the
end of this section.
A bit later we’re going to need to know S for a
uniform plane wave, so let’s work that out now. From
the plane wave relationships (Section 9.5) we find that
the magnetic field intensity associated with the
electric field in Equation 9.105 is
e = ŷ E0 e−jβz
H
η
(9.107)
p
where η = µ/ǫ is the real-valued impedance of the
medium. Let ψ be the phase of E0 ; i.e.,
E0 = |E0 |ejψ . Then
o
n
e jωt
E = Re Ee
= x̂ |E0 | cos (ωt − βz + ψ)
(9.108)
and
n
e jωt
H = Re He
o
|E0 |
= ŷ
cos (ωt − βz + ψ)
η
|E0 |2
cos2 (ωt − βz + ψ)
η
Since ω = 2πf = 2π/T , the definite integral equals
T /2. We obtain
Save =
(9.109)
(9.110)
As noted earlier, |S| is only the instantaneous power
density, which is still not quite what we are looking
|E0 |2
2η
(9.112)
It is useful to check units again at this point. Note
(V/m)2 divided by Ω is W/m2 , as expected.
Equation 9.112 is the time-average power density
(units of W/m2 ) associated with a sinusoidallyvarying uniform plane wave in lossless media.
Note that Equation 9.112 is analogous to a
well-known result from electric circuit theory. Recall
the time-average power Pave (units of W) associated
with a voltage phasor Ve across a resistance R is
Pave =
|Ve |2
2R
(9.113)
|V0+ |2
2Z0
(9.114)
which closely resembles Equation 9.112. The result is
also analogous to the result for a voltage wave on a
transmission line (Section 3.20), for which:
Pave =
Now applying Equation 9.106,
S = ẑ
for. What we are actually looking for is the
time-average power density Save – that is, the average
value of |S| over one period T of the wave. This may
be calculated as follows:
Z
1 t0 +T
|S|dt
Save =
T t=t0
Z
|E0 |2 1 t0 +T
=
cos2 (ωt − ks + ψ) dt
η T t=t0
(9.111)
where V0+ is a complex-valued constant representing
the magnitude and phase of the voltage wave, and Z0
is the characteristic impedance of the transmission
line.
Here is a good point at which to identify a common
pitfall. |E0 | and |Ve | are the peak magnitudes of the
associated real-valued physical quantities. However,
these quantities are also routinely given as root mean
square (“rms”) quantities.
√ Peak magnitudes are
greater by a factor of 2, so Equation 9.112
9.7. WAVE POWER IN A LOSSLESS MEDIUM
expressed in terms of the rms quantity lacks the factor
of 1/2.
Example 9.3. Power density of a typical radio
wave.
A radio wave transmitted from a distant location
may be perceived locally as a uniform plane
wave if there is no nearby structure to scatter the
wave; a good example of this is the wave
arriving at the user of a cellular telephone in a
rural area with no significant terrain scattering.
The range of possible signal strengths varies
widely, but a typical value of the electric field
intensity arriving at the user’s location is
10 µV/m rms. What is the corresponding power
density?
Solution: From the problem statement,
|E0 | = 10 µV/m rms. We assume propagation
occurs in air, which is indistinguishable from
free space at cellular frequencies. If we use
Equation 9.112, then we must first convert |E0 |
from rms to peak
√ magnitude, which is done by
multiplying by 2. Thus:
√
2
2 · 10 × 10−6 V/m
|E0 |2 ∼
Save =
=
2η
2 · 377 Ω
∼ 2.65 × 10−13 W/m2
=
Alternatively, we can just use a version of
Equation 9.112, which is appropriate for rms
units:
2
|E0,rms |2 ∼ 10 × 10−6 V/m
Save =
=
η
377 Ω
∼ 2.65 × 10−13 W/m2
=
Either way, we obtain the correct answer,
0.265 pW/m2 (that’s picowatts per square
meter).
Considering the prevalence of phasor representation,
it is useful to have an alternative form of the Poynting
vector which yields time-average power by operating
directly on field quantities in phasor form. This is
Save , defined as:
1 n e e ∗o
(9.115)
×H
Save , Re E
2
211
(Note that the magnetic field intensity phasor is
conjugated.) The above expression gives the expected
result for a uniform plane wave. Using
Equations 9.105 and 9.107, we find
∗ 1
E0 −jβz
−jβz
Save = Re x̂E0 e
× ŷ e
2
η
(9.116)
which yields
|E0 |2
(9.117)
Save = ẑ
2η
as expected.
Additional Reading:
• “Poynting vector” on Wikipedia.
• “Poynting’s Theorem” on Wikipedia.
[m0035]
212
CHAPTER 9. PLANE WAVES IN LOSSLESS MEDIA
Image Credits
Fig. 9.1: c Y. Qin, https://commons.wikimedia.org/wiki/File:M0142 fSphericalPhasefront.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 9.2 c Y. Qin, https://commons.wikimedia.org/wiki/File:M0142 fCylindricalPhasefront.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 9.3 c Y. Qin, https://commons.wikimedia.org/wiki/File:M0142 fPlanarPhasefront.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 9.4 c Y. Qin, https://commons.wikimedia.org/wiki/File:M0142 fPlaneWavesReflector.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 9.5 c Y. Qin, https://commons.wikimedia.org/wiki/File:M0142 fLocallyPlanar.svg,
CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
Fig. 9.7: c E. Boutet, https://en.wikipedia.org/wiki/File:Onde electromagnetique.svg,
CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Heavily modified by author.
Fig. 9.9: RJB1, https://commons.wikimedia.org/wiki/File:Linearly Polarized Wave.svg,
public domain. Modified by author.
Fig. 9.10: Dave3457, https://en.wikipedia.org/wiki/File:Circular.Polarization.Circularly.Polarized.Light
With.Components Left.Handed.svg,
public domain.
Fig. 9.11: Dave3457, https://en.wikipedia.org/wiki/File:Circular.Polarization.Circularly.Polarized.Light
Without.Components Left.Handed.svg
and https://en.wikipedia.org/wiki/File:Circular.Polarization.Circularly.Polarized.Light
Without.Components Right.Handed.svg,
public domain.
Appendix A
Constitutive Parameters of Some
Common Materials
A.1 Permittivity of Some
Common Materials
[m0135]
The values below are relative permittivity ǫr , ǫ/ǫ0
for a few materials that are commonly encountered in
electrical engineering applications, and for which
permittivity emerges as a consideration. Note that
“relative permittivity” is sometimes referred to as
dielectric constant.
Here we consider only the physical (real-valued)
permittivity, which is the real part of the complex
permittivity (typically indicated as ǫ′ or ǫ′r ) for
materials exhibiting significant loss.
Permittivity varies significantly as a function of
frequency. The values below are representative of
frequencies from a few kHz to about 1 GHz. The
values given are also representative of optical
frequencies for materials such as silica that are used
in optical applications. Permittivity also varies as a
function of temperature. In applications where
precision better than about 10% is required, primary
references accounting for frequency and temperature
should be consulted. The values presented here are
gathered from a variety of references, including those
indicated in “Additional References.”
Free Space (vacuum): ǫr , 1
Solid Dielectrics:
Material
Styrofoam1
Teflon2
Polyethylene
Polypropylene
Silica
Polystyrene
Polycarbonate
Rogers RO3003
FR4 (glass epoxy laminate)
ǫr
1.1
2.1
2.3
2.3
2.4
2.6
2.8
3.0
4.5
Common uses
coaxial cable
optical fiber3
PCB substrate
PCB substrate
1
Properly known as extruded polystyrene foam
(XPS).
2
Properly known as polytetrafluoroethylene (PTFE).
3
Typically doped with small amounts of other
materials to slightly raise or lower the index of
√
refraction (= ǫr ).
Non-conducting spacing materials used in discrete
capacitors exhibit ǫr ranging from about 5 to 50.
Semiconductors commonly appearing in electronics
– including carbon, silicon, geranium, indium
phosphide, and so on – typically exhibit ǫr in the
range 5–15.
Glass exhibits ǫr in the range 4–10, depending on
composition.
Gasses, including air, typically exhibit ǫr ∼
= 1 to
within a tiny fraction of a percent.
Liquid water typically exhibits ǫr in the range
72–81. Distilled water exhibits ǫr ≈ 81 at room
temperature, whereas sea water tends to be at the
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
214
APPENDIX A. CONSTITUTIVE PARAMETERS OF SOME COMMON MATERIALS
lower end of the range.
Other liquids typically exhibit ǫr in the range 10–90,
with considerable variation as a function of
temperature and frequency. Animal flesh and blood
consists primarily of liquid matter and so also exhibits
permittivity in this range.
Soil typically exhibits ǫr in the range 2.5–3.5 when
dry and higher when wet. The permittivity of soil
varies considerably depending on composition.
Additional Reading:
• CRC Handbook of Chemistry and Physics.
• “Relative permittivity” on Wikipedia.
• “Miscellaneous Dielectric Constants” on
microwaves101.com.
A.2 Permeability of Some
Common Materials
[m0136]
The values below are relative permeability
µr , µ/µ0 for a few materials that are commonly
encountered in electrical engineering applications,
and for which µr is significantly different from 1.
These materials are predominantly ferromagnetic
metals and (in the case of ferrites) materials
containing significant ferromagnetic metal content.
Nearly all other materials exhibit µr that is not
significantly different from that of free space.
The values presented here are gathered from a variety
of references, including those indicated in “Additional
References” at the end of this section. Be aware that
permeability may vary significantly with frequency;
values given here are applicable to the frequency
ranges for applications in which these materials are
typically used. Also be aware that materials
exhibiting high permeability are also typically
non-linear; that is, permeability depends on the
magnitude of the magnetic field. Again, values
reported here are those applicable to applications in
which these materials are typically used.
Free Space (vacuum): µr , 1.
Iron (also referred to by the chemical notation “Fe”)
appears as a principal ingredient in many materials
and alloys employed in electrical structures and
devices. Iron exhibits µr that is very high, but which
decreases with decreasing purity. 99.95% pure iron
exhibits µr ∼ 200, 000. This decreases to ∼ 5000 at
99.8% purity and is typically below 100 for purity
less than 99%.
Steel is an iron alloy that comes in many forms, with
a correspondingly broad range of permeabilites.
Electrical steel, commonly used in electrical
machinery and transformers when high permeability
is desired, exhibits µr ∼ 4000. Stainless steel,
encompassing a broad range of alloys used in
mechanical applications, exhibits µr in the range
750–1800. Carbon steel, including a broad class of
alloys commonly used in structural applications,
exhibits µr on the order of 100.
A.3. CONDUCTIVITY OF SOME COMMON MATERIALS
Ferrites include a broad range of ceramic materials
that are combined with iron and various combinations
of other metals and are used as magnets and magnetic
devices in various electrical systems. Common
ferrites exhibit µr in the range 16–640.
215
A.3 Conductivity of Some
Common Materials
[m0137]
Additional Reading:
• Section 7.16 (“Magnetic Materials”)
• CRC Handbook of Chemistry and Physics.
• “Magnetic Materials” on microwaves101.com.
• “Permeability (electromagnetism)” on
Wikipedia.
• “Iron” on Wikipedia.
• “Electrical steel” on Wikipedia.
• “Ferrite (magnet)” on Wikipedia.
The values below are conductivity σ for a few
materials that are commonly encountered in electrical
engineering applications, and for which conductivity
emerges as a consideration.
Note that materials in some applications are described
instead in terms of resistivity, which is simply the
reciprocal of conductivity.
Conductivity may vary significantly as a function of
frequency. The values below are representative of
frequencies from a few kHz to a few GHz.
Conductivity also varies as a function of temperature.
In applications where precise values are required,
primary references accounting for frequency and
temperature should be consulted. The values
presented here are gathered from a variety of
references, including those indicated in “Additional
References” at the end of this section.
Free Space (vacuum): σ , 0.
Commonly encountered elements:
Material
σ (S/m)
Copper
5.8 × 107
Gold
4.4 × 107
Aluminum 3.7 × 107
Iron
1.0 × 107
Platinum
0.9 × 107
Carbon
1.3 × 105
Silicon
4.4 × 10−4
Water exhibits σ ranging from about 6 µS/m for
highly distilled water (thus, a very poor conductor) to
about 5 S/m for seawater (thus, a relatively good
conductor), varying also with temperature and
pressure. Tap water is typically in the range
5–50 mS/m, depending on the level of impurities
present.
Soil typically exhibits σ in the range 10−4 S/m for
dry soil to about 10−1 S/m for wet soil, varying also
due to chemical composition.
216
APPENDIX A. CONSTITUTIVE PARAMETERS OF SOME COMMON MATERIALS
Non-conductors. Most other materials that are not
well-described as conductors or semiconductors and
are dry exhibit σ < 10−12 S/m. Most materials that
are considered to be insulators, including air and
common dielectrics, exhibit σ < 10−15 S/m, often by
several orders of magnitude.
Additional Reading:
• CRC Handbook of Chemistry and Physics.
• “Conductivity (electrolytic)” on Wikipedia.
• “Electrical resistivity and conductivity” on
Wikipedia.
• “Soil resistivity” on Wikipedia.
Appendix B
Mathematical Formulas
B.1 Trigonometry
B.2 Vector Operators
[m0138]
[m0139]
ejθ = cos θ + j sin θ
This section contains a summary of vector operators
expressed in each of the three major coordinate
systems:
1 jθ
e + e−jθ
2
cos θ =
sin θ =
1 jθ
e − e−jθ
j2
(B.1)
(B.2)
• cylindrical (ρ,φ,z)
(B.3)
1 1
cos θ = + cos 2θ
2 2
(B.4)
1 1
− cos 2θ
2 2
(B.5)
2
sin2 θ =
• Cartesian (x,y,z)
• spherical (r,θ,φ)
Associated basis vectors are identified using a caret (ˆ)
over the symbol. The vector operand A is expressed
in terms of components in the basis directions as
follows:
• Cartesian: A = x̂Ax + ŷAy + ẑAz
• cylindrical: A = ρ̂Aρ + φ̂Aφ + ẑAz
• spherical: A = r̂Ar + θ̂Aθ + φ̂Aφ
Gradient
Gradient in Cartesian coordinates:
∇f = x̂
∂f
∂f
∂f
+ ŷ
+ ẑ
∂x
∂y
∂z
(B.6)
Gradient in cylindrical coordinates:
∇f = ρ̂
1 ∂f
∂f
∂f
+ φ̂
+ ẑ
∂ρ
ρ ∂φ
∂z
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
(B.7)
APPENDIX B. MATHEMATICAL FORMULAS
218
Laplacian
Laplacian in Cartesian coordinates:
Gradient in spherical coordinates:
∇f = r̂
1 ∂f
1 ∂f
∂f
+ θ̂
+ φ̂
∂r
r ∂θ
r sin θ ∂φ
(B.8)
∇2 f =
Divergence
Divergence in Cartesian coordinates:
∇·A=
∂Ax
∂Ay
∂Az
+
+
∂x
∂y
∂z
(B.9)
Divergence in cylindrical coordinates:
∇·A=
1 ∂Aφ
∂Az
1 ∂
(ρAρ ) +
+
ρ ∂ρ
ρ ∂φ
∂z
(B.10)
Divergence in spherical coordinates:
1 ∂
r2 Ar
2
r ∂r
∂
1
(Aθ sin θ)
+
r sin θ ∂θ
1 ∂Aφ
+
r sin θ ∂φ
∇·A=
Curl
Curl in Cartesian coordinates:
∂Az
∂Ay
∇ × A = x̂
−
∂y
∂z
∂Ax
∂Az
+ ŷ
−
∂z
∂x
∂Ay
∂Ax
+ ẑ
−
∂x
∂y
Curl in cylindrical coordinates:
1 ∂Az
∂Aφ
∇ × A = ρ̂
−
ρ ∂φ
∂z
∂Aρ
∂Az
+ φ̂
−
∂z
∂ρ
1 ∂
∂Aρ
+ ẑ
(ρAφ ) −
ρ ∂ρ
∂φ
(B.11)
(B.12)
(B.13)
Curl in spherical coordinates:
1
∂
∂Aθ
∇ × A = r̂
(Aφ sin θ) −
r sin θ ∂θ
∂φ
1 ∂Ar
∂
1
−
(rAφ )
+ θ̂
r sin θ ∂φ
∂r
∂Ar
1 ∂
(B.14)
(rAθ ) −
+ φ̂
r ∂r
∂θ
∂2f
∂2f
∂2f
+
+
∂x2
∂y 2
∂z 2
Laplacian in cylindrical coordinates:
1 ∂
∂f
1 ∂2f
∂2f
∇2 f =
ρ
+ 2 2+ 2
ρ ∂ρ
∂ρ
ρ ∂φ
∂z
Laplacian in spherical coordinates:
∂f
1 ∂
r2
∇2 f = 2
r ∂r
∂r
1
∂ ∂f
+ 2
sin θ
r sin θ ∂θ ∂θ
1
∂2f
+ 2 2
r sin θ ∂φ2
(B.15)
(B.16)
(B.17)
B.3. VECTOR IDENTITIES
219
B.3 Vector Identities
[m0140]
Algebraic Identities
A · (B × C) = B · (C × A) = C · (A × B)
A × (B × C) = B (A · C) − C (A · B)
(B.18)
(B.19)
Identities Involving Differential Operators
∇ · (∇ × A) = 0
(B.20)
∇ × (∇f ) = 0
∇ × (f A) = f (∇ × A) + (∇f ) × A
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
2
∇ · (∇f ) = ∇ f
(B.21)
(B.22)
(B.23)
(B.24)
2
∇ × ∇ × A = ∇ (∇ · A) − ∇ A
2
∇ A = ∇ (∇ · A) − ∇ × (∇ × A)
(B.25)
(B.26)
Divergence Theorem: Given a closed surface S
enclosing a contiguous volume V,
Z
I
(∇ · A) dv =
A · ds
(B.27)
V
S
where the surface normal ds is pointing out of the
volume.
Stokes’ Theorem: Given a closed curve C bounding
a contiguous surface S,
Z
I
(∇ × A) · ds =
A · dl
(B.28)
S
C
where the direction of the surface normal ds is related
to the direction of integration along C by the “right
hand rule.”
Appendix C
Physical Constants
[m0141]
The speed of light in free space (c), which is the
phase velocity of any electromagnetic radiation in
free space, is ∼
= 2.9979 × 108 m/s. This is commonly
rounded up to 3 × 108 m/s. This rounding incurs error
of ∼
= 0.07%, which is usually much less than other
errors present in electrical engineering calculations.
The charge of an electron is ∼
= −1.602 × 10−19 C.
The constant e , +1.602176634 × 10−19 C is known
as the “elementary charge,” so the charge of the
electron is said to be −e.
The permittivity of free space (ǫ0 ) is
∼
= 8.854 × 10−12 F/m.
The permeability of free space (µ0 ) is
4π × 10−7 H/m.
The wave impedance of free space (η0 ) is the ratio
of the magnitude of the electric field intensity to that
of
p the magnetic field intensity in free space and is
µ0 /ǫ0 ∼
= 376.7 Ω. This is also sometimes referred
to as the intrinsic impedance of free space.
Electromagnetics Vol 1. c 2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1
Index
acoustic wave equation, see wave equation, acoustic
admittance, 64–66
characteristic, see characteristic admittance
air, 123, 124
aluminum, 170, 215
Ampere’s law
general form, 190–192
magnetostatics, 88, 149–150, 152, 155, 158, 159,
167
amplifier, 59, 62
antenna, 207
dipole, 207
helical, 209
impedance matching, 58, 62
monopole, 207
patch, 58
arcing, 124
attenuation constant, see propagation constant
balun, 187
Biot-Savart Law, 26
boundary conditions, 118, 120, 160, 161
capacitance, 34, 124–126
definition, 124
capacitor, 125–126
energy storage, 130
parallel plate, 18, 116–117, 126–128
carbon, 215
Cartesian coordinate system, 70, 76–77
ceramic, 170
CGS (system of units), 14
characteristic admittance, 64
characteristic impedance, 33, 37–40, 48, 63, 210
coaxial cable, 43
definition, 39
measurement, 56
charge density
line, 95, 96, 101, 112
surface, 95, 97, 112
volume, 85, 95, 99, 113
charge mobility, 136
circulation, 88
coax, see transmission line
cobalt, 171
coil, 152–154, 163, 165
toroidal, see toroidal coil
conductance, 34, 141–142
definition, 141
conductivity, 27, 136–138
definition, 136
of common materials, 215–216
conductor
good, 136, 137
perfect, 21, 118, 120, 122, 137, 144
perfect (definition), 138
poor, 123
constitutive parameters, 27
copper, 171, 215
Coulomb’s law, 93–94, 100, 103, 105
cross product, 74–75
curl, 88–89, 218
definition, 89
current, 22
conduction, 134
convection, 134
displacement, see displacement current
current density, 135–136
line, 135
surface, 135, 157
volume, 135
cylindrical coordinate system, 77–80
datum, 110
diamagnetic material, 170
dielectric (material), 42, 45, 123, 204
examples, 213
lossless, 137
lossy, 137
dielectric breakdown, 124, 138
dielectric constant, 20, 213
dielectric strength, 124
221
INDEX
222
dipole, see antenna
displacement current, 150, 192
divergence, 85–86, 218
definition, 85
divergence theorem, 87, 104, 148, 219
dot product, 73–74
duality, 146, 197
electric field, see field, electric
electric field intensity, 17–19, 94, 124
boundary conditions, 118
definition, 18
related to current, 136
electric flux density, 21–22, 85
boundary conditions, 120
definition, 21
electrical length, 53
measurement, 56
electromagnetic compatibility (EMC), 185
electromagnetic force, 19, 24
electromagnetic interference (EMI), 30, 185
electromagnetic spectrum, see spectrum
electromotive force, see emf
electron, 220
electrostatics (description), 93
emf, 179
motional, 180, 187
transformer, 180–183
emitter induction, 55
energy
electrostatic, 130–132
kinetic, 17
magnetostatic, 169–170
potential, 17, 105–107, 130
English system of units, 14
Faraday rotation, 209
Faraday’s law, 85, 176, 178–181, 189
ferrite, 170, 215
ferromagnetic material, 171
field
definition, 17
electric, 17–19, 21–22
magnetic, 22–27
scalar, 5, 17
vector, 17
field intensity, magnetic, see magnetic field intensity
field line (magnetic), 24
filter, 55
flux, 21, 24, 77, 79, 83, 85, 135, 161
definition, 85
magnetic, 163, 178
flux density, electric, see electric flux density
flux density, magnetic, see magnetic flux density
flux linkage
seelinkage, 163
force, 18, 23, 70, 105, 124
electromotive, see emf
Fourier series, 12
Fourier transform, 12
FR4, 46, 123, 213
fringing (field), 117, 126, 153, 166, 167
gamma ray, 3
Gauss’ law
electric field, 85, 100–101, 103
magnetic field, 147–148
generator, 33, 180, 187–189
geranium, 213
glass, 123, 213
Global Positioning System (GPS), 209
gold, 171, 215
gradient, 84, 114, 217
ground, 111
ground plane, 128
heat, 144
homogeneity, 27
homogeneous (media), 21, 26, 27, 196
hysteresis, 171
impedance
characteristic, see characteristic impedance
matching, 55, 62–68
of resistance, 138
wave, see wave impedance
impedance inverter, 58
independence of path, 107–108
index of refraction, 213
indium phosphide, 213
inductance, 34, 163–165
definition, 163, 164
induction, 175–178, 180
inductor, 152, 164
coaxial, 167–168
hysteresis, 171
straight coil, 165–166
infrared, 3
insulator
good, 123
INDEX
perfect, 137
poor, 137
inverse square law, 20, 25
iron, 171, 214, 215
isotropic (media), 28, 196
isotropy, 28
joule heating, 144
Joule’s law, 143
kinetic energy, see energy, kinetic
Kirchoff’s voltage law
electrostatics, 88, 105, 108–110, 118, 180, 189
Laplace’s equation, 115, 116
Laplacian (operator), 91, 218
Lenz’s law, 175, 178, 182
lightning, 124, 138
linear (media), 28, 124, 196
linear and time-invariant (LTI), 28
linearity, 28
linkage, 163
load
matched, 48
reactive, 48
lumped-element model, see transmission line
magnesium, 170
magnetic field intensity, 26–27
definition, 26
magnetic flux, see flux, magnetic
magnetic flux density, 22–25, 85
definition, 24
magnetostatics (description), 146
materials (properties), 27–28, 170–172
materials, magnetic, 170–172
Maxwell’s equations, 175, 196
differential phasor form, 194–196
source-free lossless region, 196
Maxwell-Faraday equation, 88, 180, 189–191, 202
metric system, 14
mica, 124
microstrip, 31, 44–46, 56, 63
mode, 31
monopole, see antenna
motional emf, see emf
mutual inductance, 164
neper, 40
nickel, 171
223
non-magnetic (material), 123
nonlinear (media), 27
notation, 15
Ohm’s law, 134, 136, 143, 196
ohmic heating, 144
open circuit, 48, 49, 54
optical (frequency), 3
optical fiber
multimode, 31
single mode, 32
paramagnetic material, 170
patch antenna, see antenna
perfect conductor, see conductor, perfect
permanent magnet, 22
permeability, 25–27, 123, 170
of common materials, 214–215
relative, 25, 214
permittivity, 20, 27, 124, 137
complex-valued, 197
effective, 45
of common materials, 213–214
relative, 20, 123, 213
phase velocity, 3, 8, 40, 204
coaxial cable, 43
in microstrip, 46
measurement, 57
phasefront, 198, 203
phasor, 9–13, 194, 203, 207
phasor (definition), 9
photoelectric effect, 4
photon, 19
pipe, 140
plane wave relationships, 205
platinum, 170, 215
Poisson’s equation, 91, 115–116
polarization, 206
circular, 207
elliptical, 209
linear, 206
polarizer, 207
polycarbonate, 213
polyethylene, 213
polypropylene, 213
polystyrene, 213
polytetrafluoroethylene, 213
potential difference, 106
potential energy, see energy, potential
power density, see waves
INDEX
224
power dissipation, 143–144
Poynting theorem, 210
Poynting vector, 210
printed circuit board (PCB), 44, 46, 123
capacitance, 127–128
material, 213
propagation constant, 37
attenuation, 33, 40
phase, 6, 33, 40, 197
quantum mechanics, 19, 23
quarter-wave inverter, 58
radio (frequency), 3
reactance, 165
reflection coefficient, 47–48
definition, 48
relative permeability, see permeability, relative
relative permittivity, see permittivity, relative
resistance, 34, 138–140
definition, 138
of wire (DC), 139–140
resistivity, 215
resistor, 1
RG-59, 43, 130, 142
right-hand rule
cross product, 74
magnetic field of a wire, 152, 155, 157, 182
Stokes’ theorem, 90, 151, 158, 159, 161, 163,
178, 181, 219
saturation (magnetic), 171, 186
scalar field, see field, scalar
scalar product, see dot product
semiconductor (material), 137, 213
shield, 42
short circuit, 48, 49, 54
shot noise, 94
SI (system of units), 14
signum (sgn) function, 98
silica, 213
silicon, 171, 213, 215
single-stub matching, 66–68
skin effect, 165
soil, 214, 215
solenoid, 152
sound (waves), 5
spatial frequency, 6
spectrum, 3
speed of light, 3, 204, 220
spherical coordinate system, 81–83
standing wave, 41, 49–50
definition, 49
standing wave ratio (SWR), 51–52
definition, 51
steel, 140, 214
Stokes’ theorem, 90, 109, 163, 178, 181, 189, 190,
219
stripline, 44
stub
applications, 55, 66–68
definition, 54
open circuit, 54
short circuit, 54
styrofoam, 213
superposition, 9, 28
susceptance, 64, 67
symmetry, 100, 102, 151
teflon, 123, 213
telegrapher’s equations, 35–37
TEM line, see transmission line
Thévenin equivalent circuit, 33
time-invariance (media), 28
toroid, 155
toroidal coil, 155–157
transformer, 164, 183–187
emf, see emf
hysteresis, 171
induction, 176
transmission line
coaxial, 31, 42–44, 62, 130, 141–142
definition, 30
input impedance, 52–55
lossless, 40
low loss, 40, 41, 43
lumped-element model, 34–35, 41
microstrip, see microstrip
quarter-wavelength, 57–60
stripline, 44
stub, see stub
time-average power, 61
transverse electromagnetic (TEM), 31, 37, 42, 45
two-port representation, 33–34
wave equations, 37
two-port, 33
ultraviolet, 3
unit vector, 70
units, 13–14
INDEX
vector
arithmetic, 70–75, 219
definition, 70
identity, 219
position-fixed, 70
position-free, 70
unit, 70
velocity, 70
voltage reflection coefficient, see reflection coefficient
voltage standing wave ratio (VSWR), 51
water, 213, 215
wave equation
acoustic, 6
electromagnetic, 91
source-free lossless region, 196–197
TEM transmission line, 37
wave impedance, 197, 202, 220
waveguide, 31
wavelength, 2, 3, 7, 204
in microstrip, 46
wavenumber, 6
waves
cylindrical, 198
definition, 17
fundamentals, 5–8
guided, 9, 30
plane, 198–206, 209
polarization, 206–209
power density, 209–211
spherical, 198
standing, see standing wave
transmission line analogy, 210
unguided, 9
wire
DC resistance of, 139–140
work, 105
X-ray, 3
225